Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear...
Transcript of Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear...
![Page 1: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/1.jpg)
Linear and Nonlinear EquationsModeling
Ordinary Differential Equations. Session 2
Dr. Marco A Roque Sol
17/01/2017
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 2: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/2.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 3: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/3.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t)
are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 4: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/4.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β)
that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 5: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/5.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0.
Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 6: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/6.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0,
there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 7: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/7.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t)
on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 8: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/8.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β)
to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 9: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/9.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 10: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/10.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 11: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/11.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 12: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/12.jpg)
Linear and Nonlinear Equations
Theorem 1 (Existence and Uniqueness).
Suppose p(t) and g(t) are continuous real-valued functions on aninterval (α, β) that contains the point t0. Then, for any choice of(initial value) y0, there exists a unique solution y(t) on the wholeinterval (α, β) to the linear differential equation
dy(t)
dt+ p(t)y(t) = g(t)
for all t ∈ (α, β) and y(t0) = yo.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 13: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/13.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 14: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/14.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y)
and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 15: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/15.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y)
are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 16: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/16.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 17: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/17.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 18: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/18.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0).
Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 19: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/19.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ)
so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 20: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/20.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 21: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/21.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 22: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/22.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution
for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 23: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/23.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 24: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/24.jpg)
Linear and Nonlinear Equations
Theorem 2 (Existence and Uniqueness).
Suppose that both f (t, y) and∂f
∂y(t, y) are continuous functions
defined on a region R as
R = {(t, y) : t0 − δ < t < t0 + δ; y0 − ε < y < y0 + ε}
containing the point (t0, y0). Then there exists a number δ1(possibly smaller than δ) so that a solution y = f (t) to
y ′ = f (t, y) y(t0) = y0
is the unique solution for t0 − δ1 < t < t0 + δ1
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 25: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/25.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 26: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/26.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 27: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/27.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 28: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/28.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case,
both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 29: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/29.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1
and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 30: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/30.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1
are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 31: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/31.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y).
The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 32: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/32.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees
that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 33: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/33.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists
in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 34: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/34.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1,
and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 35: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/35.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique
in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 36: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/36.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 37: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/37.jpg)
Linear and Nonlinear Equations
Example 21
Consider the ODE
y ′ = t − y + 1; y(1) = 2
In this case, both the function f (t, y) = ty + 1 and its partial
derivative∂f
∂y(t, y) = −1 are defined and continuous at all points
(x , y). The Theorem 2 guarantees that a solution to the ODEexists in some open interval centered at 1, and that this solution isunique in some (possibly smaller) interval centered at 1.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 38: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/38.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 39: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/39.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 40: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/40.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t.
In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 41: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/41.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words,
in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 42: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/42.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1
as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 43: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/43.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 44: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/44.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 45: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/45.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 46: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/46.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 47: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/47.jpg)
Linear and Nonlinear Equations
In fact, an explicit solution to this equation is
y(t) = t + e1t
This solution exists (and is the unique solution to the equation) forall real numbers t. In other words, in this example we may choosethe numbers δ and δ1 as large as we please.
Example 22
Consider the ODE
y ′ = 1 + y2; y(0) = 0
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 48: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/48.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2
and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 49: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/49.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y
are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 50: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/50.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous
at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 51: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/51.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 52: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/52.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees
that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 53: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/53.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists
insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 54: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/54.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0,
and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 55: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/55.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is unique
in some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 56: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/56.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 57: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/57.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating,
we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 58: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/58.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 59: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/59.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 60: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/60.jpg)
Linear and Nonlinear Equations
In this case, both the function f (t, y) = 1 + y2 and its partial
derivative∂f
∂y(t, y) = 2y are defined and continuous at all points
(t, y).
The Theorem 2 guarantees that a solution to the ODE exists insome open interval centered at 0, and that this solution is uniquein some (possibly smaller) interval centered at 1.
By separating variables and integrating, we derive a solution to thisequation of the form
y(t) = tan(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 61: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/61.jpg)
Linear and Nonlinear Equations
As an abstract function of t,
this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 62: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/62.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ...
However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 63: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/63.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain.
Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 64: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/64.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 65: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/65.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 66: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/66.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE.
In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 67: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/67.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2,
although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 68: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/68.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ,
may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 69: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/69.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 70: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/70.jpg)
Linear and Nonlinear Equations
As an abstract function of t, this is defined for allt 6= ...,−3π/2,−π/2, π/2, 3π/2, ... However, in order for thisfunction to be considered as a solution to this ODE, we mustrestrict the domain. Specifically, the function
y(t) = tan(t); −π/2 < x < π/2
is a solution to the above ODE. In this example we must chooseδ1 = π/2, although the initial value δ, may be chosen as large aswe please.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 71: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/71.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 72: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/72.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation.
Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 73: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/73.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations
that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 74: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/74.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job
are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 75: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/75.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time,
modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 76: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/76.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation
that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 77: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/77.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 78: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/78.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling
and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 79: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/79.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 80: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/80.jpg)
Modeling
We now move into one of the main applications of differentialequations.
Modeling is the process of writing a differential equation todescribe a physical situation. Almost all of the differentialequations that you will use in your job are there becausesomebody, at some time, modeled a situation to come up with thedifferential equation that you are using.
This section is designed to introduce you to the process ofmodeling and show you what is involved in modeling.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 81: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/81.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 82: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/82.jpg)
Modeling
Mixing Problems
In these problems we will start
with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 83: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/83.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid.
Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 84: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/84.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 85: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/85.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank
may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 86: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/86.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not
contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 87: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/87.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it.
Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 88: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/88.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank
will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 89: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/89.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 90: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/90.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t)
gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 91: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/91.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance
dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 92: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/92.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t
we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 93: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/93.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved,
will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 94: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/94.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 95: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/95.jpg)
Modeling
Mixing Problems
In these problems we will start with a substance that is dissolved ina liquid. Liquid will be entering and leaving a holding tank.
The liquid entering the tank may or may not contain more of thesubstance dissolved in it. Liquid leaving the tank will of coursecontain the substance dissolved in it.
If Q(t) gives the amount of the substance dissolved in the liquid inthe tank at any time t we want to develop a differential equationthat, when solved, will give us an expression for Q(t) .
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 96: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/96.jpg)
Modeling
Note as well that in many situations
we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 97: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/97.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions
and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 98: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/98.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid,
but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 99: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/99.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 100: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/100.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here
is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 101: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/101.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid
is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 102: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/102.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank.
Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 103: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/103.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case,
but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 104: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/104.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank,
theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 105: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/105.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 106: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/106.jpg)
Modeling
Note as well that in many situations we can think of air as a liquidfor the purposes of these kinds of discussions and so we dontactually need to have an actual liquid, but could instead use air asthe liquid.
The main assumption that well be using here is that theconcentration of the substance in the liquid is uniform throughoutthe tank. Clearly this will not be the case, but if we allow theconcentration to vary depending on the location in the tank, theproblem becomes very difficult.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 107: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/107.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 108: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/108.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 109: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/109.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 110: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/110.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 111: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/111.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 112: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/112.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 113: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/113.jpg)
Modeling
The main equation that well be using to model this situation is :
Rate of change of Q(t) =
Rate at which Q(t) enters the tank -
Rate at which Q(t) exits the tank
where,
Rate of change of Q(t) =dQ(t)
dt= Q(t)′
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 114: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/114.jpg)
Modeling
Rate at which Q(t) enters the tank = (flow rate of liquid entering)x (concentration of substance in liquid entering)
Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x(concentration of substance in liquid exiting)
Let’s take a look at the first problem
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 115: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/115.jpg)
Modeling
Rate at which Q(t) enters the tank = (flow rate of liquid entering)x (concentration of substance in liquid entering)
Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x(concentration of substance in liquid exiting)
Let’s take a look at the first problem
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 116: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/116.jpg)
Modeling
Rate at which Q(t) enters the tank = (flow rate of liquid entering)x (concentration of substance in liquid entering)
Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x(concentration of substance in liquid exiting)
Let’s take a look at the first problem
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 117: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/117.jpg)
Modeling
Rate at which Q(t) enters the tank = (flow rate of liquid entering)x (concentration of substance in liquid entering)
Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x(concentration of substance in liquid exiting)
Let’s take a look at the first problem
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 118: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/118.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it.
Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 119: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/119.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank
has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 120: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/120.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal.
If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 121: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/121.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr,
how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 122: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/122.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 123: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/123.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water
enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 124: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/124.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank
to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 125: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/125.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point.
Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 126: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/126.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality,
but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 127: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/127.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 128: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/128.jpg)
Modeling
Example 14
A 1500 gallon tank initially contains 600 gallons of water with 5lbs of salt dissolved in it. Water enters the tank at a rate of 9gal/hr and the water entering the tank has a salt concentration of15 (1 + cos(t)) lbs/gal. If a well mixed solution leaves the tank at arate of 6 gal/hr, how much salt is in the tank when it overflows?
We are going to assume that the instant the water enters the tankit somehow instantly disperses evenly throughout the tank to givea uniform concentration of salt in the tank at every point. Again,this will clearly not be the case in reality, but it will allow us to dothe problem.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 129: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/129.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 130: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/130.jpg)
Modeling
Solution
Now,
to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 131: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/131.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t)
wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 132: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/132.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering,
the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 133: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/133.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering,
the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 134: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/134.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and
the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 135: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/135.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 136: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/136.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tank
the concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 137: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/137.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank
and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 138: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/138.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 139: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/139.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 140: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/140.jpg)
Modeling
Solution
Now, to set up the IVP that we will need to solve to get Q(t) wewill need the flow rate of the water entering, the concentration ofthe salt in the water entering, the flow rate of the water leaving(weve got that) and the concentration of the salt in the waterexiting (we dont have this )
Since we are assuming a uniform concentration of salt in the tankthe concentration at any point in the tank and hence in the waterexiting is given by,
Concentration = (Amount of salt in the tank at any time t) / (Volume of water in the tank at any time t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 141: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/141.jpg)
Modeling
The amount at any time t is easy it’s just Q(t).
The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 142: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/142.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy.
We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 143: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/143.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons
and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 144: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/144.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters
and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 145: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/145.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave.
So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 146: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/146.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours,
everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 147: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/147.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank,
or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 148: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/148.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t
there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 149: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/149.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 150: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/150.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 151: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/151.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 152: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/152.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 153: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/153.jpg)
Modeling
The amount at any time t is easy it’s just Q(t). The volume isalso pretty easy. We start with 600 gallons and every hour 9gallons enters and 6 gallons leave. So, if we use t in hours, everyhour 3 gallons enters the tank, or at any time t there is 600 + 3tgallons of water in the tank.
So, the IVP for this situation is,
Q ′(t) = (9)
(1
5(1 + cos(t))
)− 6
(Q(t)
600 + 3t
)Q(0) = 5
Q ′(t) =9
5(1 + cos(t))− 2
(Q(t)
200 + t
)Q(0) = 5
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 154: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/154.jpg)
Modeling
This is a linear differential equation.
We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 155: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/155.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 156: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/156.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 157: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/157.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 158: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/158.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now,
multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 159: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/159.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equation
by the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 160: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/160.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 161: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/161.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 162: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/162.jpg)
Modeling
This is a linear differential equation. We will show most of thedetails, but leave the description of the solution process out.
Q ′(t) +
(2
Q(t)
200 + t
)=
9
5(1 + cos(t))
Now find the integrating factor:
µ(t) = e∫
2200+t
dt = e2ln(200+t) = e ln(200+t)2 = (200 + t)2
Now, multiply the rewritten differential equationby the integratingfactor.
(200 + t)2Q ′(t) + (200 + t)2(
2Q(t)
200 + t
)= (200 + t)2
9
5(1 + cos(t))
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 163: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/163.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 164: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/164.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.
∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 165: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/165.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 166: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/166.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 167: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/167.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 168: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/168.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 169: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/169.jpg)
Modeling
((200 + t)2Q(t)
)′= (200 + t)2
9
5(1 + cos(t))
Integrate both sides and solve for the solution.∫ ((200 + t)2Q(t)
)′dt =
∫9
5(200 + t)2 (1 + cos(t)) dt
(200 + t)2Q(t) =9
5[1
3(200 + t)3 + (200 + t)2sin(t) + ...
...+ 2(200 + t)cos(t)− 2sin(t)] + c
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)+
c
(200 + t)2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 170: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/170.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 171: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/171.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 172: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/172.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 173: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/173.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 174: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/174.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 175: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/175.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 176: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/176.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 177: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/177.jpg)
Modeling
And applying initial conditions
5 = Q(0) =9
5
(1
3(200) +
2
200
)+
c
(200)2
c = −4600720
The solution is then,
Q(t) =9
5
(1
3(200 + t)2 + sin(t) +
2cos(t)
200 + t− 2sin(t)
(200 + t)2
)− 4600720
(200 + t)2
Now, the tank will overflow at t = 300hrs. The amount of salt inthe tank at that time is.
Q(300) = 279.80lbs
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 178: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/178.jpg)
Modeling
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 179: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/179.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 180: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/180.jpg)
Modeling
Example 15
A 1000 gallon holding tank
that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 181: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/181.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water
with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 182: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/182.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it.
Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 183: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/183.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr
and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 184: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/184.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it.
A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 185: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/185.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution
leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 186: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/186.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well.
When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 187: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/187.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off
and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 188: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/188.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr
while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 189: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/189.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.
Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 190: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/190.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 191: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/191.jpg)
Modeling
Example 15
A 1000 gallon holding tank that catches runoff from some chemicalprocess initially has 800 gallons of water with 2 ounces of pollutiondissolved in it. Polluted water flows into the tank at a rate of 3gal/hr and contains 5 ounces/gal of pollution in it. A well mixedsolution leaves the tank at 3 gal/hr as well. When the amount ofpollution in the holding tank reaches 500 ounces the inflow ofpolluted water is cut off and fresh water will enter the tank at adecreased rate of 2 gal/hr while the outflow is increased to 4gal/hr.Determine the amount of pollution in the tank at any time t.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 192: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/192.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 193: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/193.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes.
If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 194: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/194.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed
there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 195: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/195.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 196: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/196.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well.
In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 197: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/197.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem.
One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 198: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/198.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation
whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 199: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/199.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank
and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 200: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/200.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached
and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 201: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/201.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 202: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/202.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 203: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/203.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 204: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/204.jpg)
Modeling
Solution
The pollution in the tank will increase as time passes. If theamount of pollution ever reaches the maximum allowed there willbe a change in the situation.
This will necessitate a change in the differential equationdescribing the process as well. In other words, we’ll need two IVP’sfor this problem. One will describe the initial situation whenpolluted runoff is entering the tank and one for after the maximumallowed pollution is reached and fresh water is entering the tank.
Here are the two IVPs for this problem.
Q ′1(t) = (3)(5)− 3
(Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 205: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/205.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 206: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/206.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward
and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 207: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/207.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached.
We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 208: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/208.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.
Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 209: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/209.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time
sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 210: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/210.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy
with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 211: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/211.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term
as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 212: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/212.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 213: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/213.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one.
First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 214: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/214.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0.
We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 215: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/215.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm,
thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 216: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/216.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts.
Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 217: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/217.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank
and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 218: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/218.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero.
This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 219: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/219.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term,
and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 220: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/220.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 221: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/221.jpg)
Modeling
Q ′2(t) = (2)(0)− 4
(Q2(t)
800− 2(t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first one is fairly straight forward and will be valid until themaximum amount of pollution is reached. We’ll call that time tm.Also, the volume in the tank remains constant during this time sowe dont need to do anything fancy with that this time in thesecond term as we did in the previous example.
We will need a little explanation for the second one. First noticethat we do not start over at t = 0. We start this one at tm, thetime at which the new process starts. Next, fresh water is flowinginto the tank and so the concentration of pollution in the incomingwater is zero. This will drop out the first term, and thats okay sodon’t worry about that.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 222: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/222.jpg)
Modeling
Now,
notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 223: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/223.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.
During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 224: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/224.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process
so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 225: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/225.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat.
However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 226: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/226.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample.
When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 227: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/227.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts up
there needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 228: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/228.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank
and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 229: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/229.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons
that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 230: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/230.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation.
So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 231: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/231.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume
we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 232: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/232.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times.
In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 233: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/233.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1)
we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 234: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/234.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons
in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 235: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/235.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 236: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/236.jpg)
Modeling
Now, notice that the volume at any time looks a little funny.During this time frame we are losing two gallons of water everyhour of the process so we need the − 2 in there to account forthat. However, we cant just use t as we did in the previousexample. When this new process starts upthere needs to be 800gallons of water in the tank and if we just use t there we won’thave the required 800 gallons that we need in the equation. So, tomake sure that we have the proper volume we need to put in thedifference in times. In this way once we are one hour into the newprocess (i.e t - tm = 1) we will have 798 gallons in the tank asrequired.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 237: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/237.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty.
This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 238: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/238.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te .
Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 239: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/239.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400
since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 240: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/240.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty
400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 241: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/241.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 242: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/242.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well,
it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 243: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/243.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along
and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 244: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/244.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 245: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/245.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay,
now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 246: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/246.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of
here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 247: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/247.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 248: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/248.jpg)
Modeling
Finally, the second process can’t continue forever as eventually thetank will empty. This is denoted in the time restrictions as te . Wecan also note that te = tm + 400 since the tank will empty 400hours after this new process starts up.
Well, it will end provided something doesn’t come along and startchanging the situation again.
Okay, now that we’ve got all the explanations taken care of here’sthe simplified version of the IVP’s that we’ll be solving.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 249: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/249.jpg)
Modeling
Q ′1(t) = 15−(
3Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Q ′2(t) = −(
2Q2(t)
400− (t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first IVP is a fairly simple linear differential equation so we willleave the details of the solution to you to check.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 250: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/250.jpg)
Modeling
Q ′1(t) = 15−(
3Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Q ′2(t) = −(
2Q2(t)
400− (t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first IVP is a fairly simple linear differential equation so we willleave the details of the solution to you to check.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 251: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/251.jpg)
Modeling
Q ′1(t) = 15−(
3Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Q ′2(t) = −(
2Q2(t)
400− (t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first IVP is a fairly simple linear differential equation
so we willleave the details of the solution to you to check.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 252: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/252.jpg)
Modeling
Q ′1(t) = 15−(
3Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Q ′2(t) = −(
2Q2(t)
400− (t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first IVP is a fairly simple linear differential equation so we willleave the details of the solution to you to check.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 253: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/253.jpg)
Modeling
Q ′1(t) = 15−(
3Q1(t)
800
)Q(0) = 2 0 ≤ t ≤ tm
Q ′2(t) = −(
2Q2(t)
400− (t − tm)
)Q(tm) = 500 tm ≤ t ≤ te
The first IVP is a fairly simple linear differential equation so we willleave the details of the solution to you to check.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 254: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/254.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 255: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/255.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now,
we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 256: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/256.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm.
We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 257: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/257.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500.
So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 258: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/258.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 259: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/259.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 260: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/260.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 261: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/261.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours.
Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 262: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/262.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake
here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 263: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/263.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 264: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/264.jpg)
Modeling
Q1(t) = 4000− 3998e−3t800
Now, we need to find tm. We need to do is determine when theamount of pollution reaches 500. So we need to solve.
Q1(t) = 4000− 3998e−3t800 = 500
tm = 35.475
So, the second process will pick up at 35.475 hours. Forcompleteness sake here is the IVP with this information inserted
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 265: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/265.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 266: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/266.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both
linear and separable and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 267: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/267.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable
and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 268: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/268.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable and I willleave the details to you again
to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 269: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/269.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 270: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/270.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 271: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/271.jpg)
Modeling
Q ′2(t) = −(
2Q2(t)
435.475− t
)Q(35.475) = 50035.475 ≤ t ≤ 435.475
This differential equation is both linear and separable and I willleave the details to you again to check that we should get.
Q2(t) =(435.476− t)2
320
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 272: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/272.jpg)
Modeling
So, a solution that encompasses the complete running time of theprocess is
Q(t) =
{4000− 3998e−
3t800 0 ≤ t ≤ 35.475
(435.476−t)2320 35.475 ≤ t ≤ 435.4758
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 273: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/273.jpg)
Modeling
So, a solution that encompasses the complete running time of theprocess is
Q(t) =
{4000− 3998e−
3t800 0 ≤ t ≤ 35.475
(435.476−t)2320 35.475 ≤ t ≤ 435.4758
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 274: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/274.jpg)
Modeling
So, a solution that encompasses the complete running time of theprocess is
Q(t) =
{4000− 3998e−
3t800 0 ≤ t ≤ 35.475
(435.476−t)2320 35.475 ≤ t ≤ 435.4758
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 275: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/275.jpg)
Modeling
Here is graph of the solution.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 276: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/276.jpg)
Modeling
Here is graph of the solution.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 277: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/277.jpg)
Modeling
Here is graph of the solution.
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 278: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/278.jpg)
Modeling
Example 16
Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 279: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/279.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 280: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/280.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation
only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 281: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/281.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth,
and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 282: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/282.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative
to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 283: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/283.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth.
If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 284: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/284.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances,
then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 285: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/285.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity
is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 286: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/286.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.
Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 287: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/287.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation
tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 288: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/288.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r
from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 289: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/289.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 290: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/290.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 291: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/291.jpg)
Modeling
Example 16Escape Velocity
The model of constant gravitation only works when were close tothe surface of the earth, and the distances we’re dealing with aresmall relative to the radius of the earth. If we start to deal withlarger distances, then we must take into account that accelerationfrom gravity is weaker the farther we are away from the earth.Newton’s law of universal gravitation tells us that the force fromgravity experienced a distance r from the center of the earth willbe:
F = −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 292: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/292.jpg)
Modeling
where m is the mass of the object,
M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 293: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/293.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth,
andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 294: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/294.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2.
Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 295: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/295.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth.
This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 296: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/296.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed
at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 297: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/297.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface
if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 298: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/298.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth
andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 299: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/299.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 300: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/300.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well,
we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 301: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/301.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth
along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 302: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/302.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center,
then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 303: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/303.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 304: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/304.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 305: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/305.jpg)
Modeling
where m is the mass of the object, M is the mass of the earth, andG is Newtons gravitational constant G = 6.6710−11Nm2/kg2. Wecan use this relation to calculate an objects escape velocity onthe surface of the earth. This is the speed at which an object mustbe moving away from the earth at the earths surface if it is tobreak free from the gravitational attraction of the earth andcontinue to move away forever.
Well, we note that if we move away from the earth along a line thatgoes through the earths center, then Newtons second law tells us:
md2r
dt2= −GmM
r2
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 306: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/306.jpg)
Modeling
By the chain rule we have
dvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 307: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/307.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 308: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/308.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 309: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/309.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 310: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/310.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 311: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/311.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 312: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/312.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 313: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/313.jpg)
Modeling
By the chain rule we havedvdt = dv
drdrdt , and if we note v = dr
dt then we can transform thisrelation into
mvdv
dr= −GmM
r2
If we integrate both sides with respect to r we get:
1
2mv2 =
GmM
r+ c
And applying initial conditions, v(R) = v0, we obtain:
v2 = v20 + 2GM
(1
r− 1
R
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 314: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/314.jpg)
Modeling
If the object is to escape from the graviational action of the earth,
then its velocity must always be positive as r →∞. This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 315: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/315.jpg)
Modeling
If the object is to escape from the graviational action of the earth,then its velocity must always be positive as r →∞.
This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 316: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/316.jpg)
Modeling
If the object is to escape from the graviational action of the earth,then its velocity must always be positive as r →∞. This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 317: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/317.jpg)
Modeling
If the object is to escape from the graviational action of the earth,then its velocity must always be positive as r →∞. This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 318: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/318.jpg)
Modeling
If the object is to escape from the graviational action of the earth,then its velocity must always be positive as r →∞. This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 319: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/319.jpg)
Modeling
If the object is to escape from the graviational action of the earth,then its velocity must always be positive as r →∞. This will bethe case if
v0 ≥√
2GM
R
For the earth the escape velocity is v0 = 11, 180m/s
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 320: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/320.jpg)
Modeling
Example 17
Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 321: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/321.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 322: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/322.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles,
with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 323: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/323.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles.
If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 324: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/324.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth
while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 325: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/325.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit,
can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 326: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/326.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid
using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 327: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/327.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 328: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/328.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 329: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/329.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 330: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/330.jpg)
Modeling
Example 17Escape Velocity
Suppose that you are stranded -your rocket engine has failed- onan asteroid of diameter 3 miles, with density equal to that of theearth with radius 3960 miles. If you have enough spring in yourlegs to jump 4 feet straight up on earth while wearing your spacesuit, can you blast off from this asteroid using leg power alone ?
Solution
The escape velocity for the Earth is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 331: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/331.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 332: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/332.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 333: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/333.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 334: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/334.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 335: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/335.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 336: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/336.jpg)
Modeling
vE =
√2GME
RE
Solving for ME in this equation we get:
ME =v2ERE
2G
The density of the earth is its mass divided by its volume
ME43πR
3E
=3v2E
8πGR2E
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 337: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/337.jpg)
Modeling
A similar calculation can be done for the asteroid,
and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 338: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/338.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 339: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/339.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 340: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/340.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 341: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/341.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 342: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/342.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 343: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/343.jpg)
Modeling
A similar calculation can be done for the asteroid, and given boththe asteroid and the Earth have the same density we get:
3v2E8πGR2
E
=3v2A
8πGR2A
With a little algebra from this we can deduce the ratio:
v2Av2E
=R2A
R2E
So, the escape velocity from the asteroid is
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 344: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/344.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 345: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/345.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump,
all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 346: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/346.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2,
and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 347: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/347.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy
is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 348: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/348.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh.
So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 349: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/349.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 350: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/350.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 351: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/351.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 352: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/352.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 353: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/353.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 354: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/354.jpg)
Modeling
vA = vE
(RA
RE
)= 11, 180
(1.5
3960
)= 4.24m/s
At the begining of the jump, all the energy is initially kinetic,12mv2, and at the top of the jump all that energy is converted intopotential energy, mgh. So, the final height is given by the equation:
v =√
2gh
Plugging 4 feet in for h we get:
v =√
2(9.8)(4ft)(1m/3.28ft) = 4.89m/s > 4.24m/s
So, yes, you can get off the asteroid !!!!
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 355: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/355.jpg)
Modeling
Example 18
Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 356: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/356.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 357: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/357.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money.
Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 358: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/358.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.
We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 359: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/359.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations.
Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 360: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/360.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously.
If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 361: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/361.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,
then the rate of change of the initial deposit is dSdt , this quantity is
equal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 362: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/362.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt ,
this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 363: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/363.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues,
which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 364: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/364.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times
the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 365: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/365.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t).
Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 366: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/366.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 367: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/367.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 368: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/368.jpg)
Modeling
Example 18Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.We can model the growth of an initial deposit with respect to theinterest rate r with differential equations. Let’s assume that theinitial deposit is compounded continuously. If t represents time,then the rate of change of the initial deposit is dS
dt , this quantity isequal to the rate at which interest accrues, which is the interestrate r times the current value of the investment S(t). Thus, themodel is given by
dS
dt= rS(t)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 369: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/369.jpg)
Modeling
Suppose
that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 370: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/370.jpg)
Modeling
Suppose that we also know the value of the investment
at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 371: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/371.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 372: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/372.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 373: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/373.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 374: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/374.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 375: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/375.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 376: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/376.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt
∫1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 377: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/377.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 378: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/378.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 379: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/379.jpg)
Modeling
Suppose that we also know the value of the investment at someparticular time, say,
S(0) = S0
Integrating this separable IVP
dS
dt= rS(t)
1
SdS = rdt∫
1
SdS =
∫rdt
ln|S | = rt + K
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 380: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/380.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0. Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 381: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/381.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0,
we have that theconstant of integration is c = S0. Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 382: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/382.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0.
Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 383: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/383.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0. Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 384: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/384.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0. Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 385: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/385.jpg)
Modeling
S(t) = cert
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0. Therefore the solution to thisinitial value problem is:
S(t) = S0ert
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 386: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/386.jpg)
Modeling
Example 19
Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 387: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/387.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 388: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/388.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money.
Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 389: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/389.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.
Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 390: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/390.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously.
Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 391: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/391.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest,
dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 392: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/392.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k.
So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 393: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/393.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 394: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/394.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 395: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/395.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 396: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/396.jpg)
Modeling
Example 19Compound Interest
Let S be an initial sum of money. Let r represent an interest rate.Assume that the initial deposit is compounded continuously. Let’suppose that there may be deposits or withdrawals in addition tothe accrual of interest, dividends, or capital gains and the depositsor withdrawals take place at a constant rate k. So the IVP is
dS
dt= rS(t) + k S(0) = S0
( k > 0 for deposits and k < 0 for withdrawals)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 397: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/397.jpg)
Modeling
The above differential equation is a linear one,
with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 398: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/398.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt ,
so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 399: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/399.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 400: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/400.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 401: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/401.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0,
we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 402: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/402.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r .
Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 403: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/403.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 404: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/404.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 405: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/405.jpg)
Modeling
The above differential equation is a linear one, with integratingfactor e−rt , so its general solution is:
S(t) = cert − k
r
and using the initial condition that S(0) = S0, we have that theconstant of integration is c = S0 + k
r . Therefore the solution tothis initial value problem is:
S(t) = S0ert +
k
r
(ert − 1
)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 406: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/406.jpg)
Modeling
Example 20
Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 407: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/407.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 408: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/408.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states
that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 409: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/409.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a body
changes at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 410: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/410.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperature
between its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 411: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/411.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature
and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 412: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/412.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 413: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/413.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 414: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/414.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 415: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/415.jpg)
Modeling
Example 20Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of a bodychanges at a rate proportional to the difference in temperaturebetween its own temperature and the temperature of itssurroundings.
We can therefore write
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 416: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/416.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 417: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/417.jpg)
Modeling
where,
T = Temperature of the body at any time t
Ts= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 418: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/418.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)
T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 419: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/419.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the body
k = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 420: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/420.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 421: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/421.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 422: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/422.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 423: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/423.jpg)
Modeling
where,
T = Temperature of the body at any time tTs= Temperature of the surroundings (also called ambienttemperature)T0 = Initial temperature of the bodyk = constant of proportionality
Integrating this separable differential equation
dT
dt= −k (T − Ts)
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 424: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/424.jpg)
Modeling
1
T − TsdT = −kdt
∫1
T − TsdT = −
∫kdt
ln|T − Ts | = −kt + C
|T − Ts | = e−kt+C
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 425: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/425.jpg)
Modeling
1
T − TsdT = −kdt∫
1
T − TsdT = −
∫kdt
ln|T − Ts | = −kt + C
|T − Ts | = e−kt+C
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 426: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/426.jpg)
Modeling
1
T − TsdT = −kdt∫
1
T − TsdT = −
∫kdt
ln|T − Ts | = −kt + C
|T − Ts | = e−kt+C
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 427: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/427.jpg)
Modeling
1
T − TsdT = −kdt∫
1
T − TsdT = −
∫kdt
ln|T − Ts | = −kt + C
|T − Ts | = e−kt+C
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 428: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/428.jpg)
Modeling
1
T − TsdT = −kdt∫
1
T − TsdT = −
∫kdt
ln|T − Ts | = −kt + C
|T − Ts | = e−kt+C
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 429: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/429.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 430: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/430.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 431: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/431.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 432: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/432.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 433: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/433.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2
![Page 434: Ordinary Di erential Equations. Session 2roquesol/Math_308_Fall_2017... · 2017-09-12 · Linear and Nonlinear Equations Theorem 1 (Existence and Uniqueness). Suppose p(t) and g(t)](https://reader031.fdocuments.net/reader031/viewer/2022011823/5ecf4bd2ef0e483c994b6b1c/html5/thumbnails/434.jpg)
Modeling
T = Ts + ce−kt ; T > Ts (why?)
Applying initial conditions T (0) = T0
c = T0 − Ts
Thus, the solution is
T = Ts + (T0 − Ts)e−kt
Dr. Marco A Roque Sol Ordinary Differential Equations. Session 2