Post on 11-Mar-2020
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Definition An Example Double Check
Separable Differential Equations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
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Definition An Example Double Check
What are Separable Differential Equations?
1. A separable differential equation is of the formy′ = f (x)g(y).
2. That is, a differential equation is separable if the terms thatare not equal to y′ can be factored into a factor that onlydepends on x and another factor that only depends on y.
3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use
integrals to undo the differentials dx and dy from y′ =dydx
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
What are Separable Differential Equations?1. A separable differential equation is of the form
y′ = f (x)g(y).
2. That is, a differential equation is separable if the terms thatare not equal to y′ can be factored into a factor that onlydepends on x and another factor that only depends on y.
3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use
integrals to undo the differentials dx and dy from y′ =dydx
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
What are Separable Differential Equations?1. A separable differential equation is of the form
y′ = f (x)g(y).2. That is, a differential equation is separable if the terms that
are not equal to y′ can be factored into a factor that onlydepends on x and another factor that only depends on y.
3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use
integrals to undo the differentials dx and dy from y′ =dydx
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
What are Separable Differential Equations?1. A separable differential equation is of the form
y′ = f (x)g(y).2. That is, a differential equation is separable if the terms that
are not equal to y′ can be factored into a factor that onlydepends on x and another factor that only depends on y.
3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use
integrals to undo the differentials dx and dy from y′ =dydx
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
dydx
= xey
dyey = x dx
∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey
dyey = x dx
∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey
dyey = x dx
∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx
∫e−y dy =
∫x dx
−e−y =12
x2 + c
e−y = −c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = −c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = −c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = −c− 12
x2
−y = ln(
c− 12
x2)
y = − ln(
c− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = C− 12
x2
−y = ln(
C− 12
x2)
y = − ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = C− 12
x2
−y = ln(
C− 12
x2)
y = − ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.dydx
= xey∫ dyey =
∫x dx∫
e−y dy =∫
x dx
−e−y =12
x2 + c
e−y = C− 12
x2
−y = ln(
C− 12
x2)
y = − ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
The general solution is a family of functions.
y =− ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
The general solution is a family of functions.
y =− ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
The general solution is a family of functions.
y =− ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
The general solution is a family of functions.
y =− ln(
C− 12
x2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?We don’t.
I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?
Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?We don’t.
I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?We don’t.
I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?
We don’t.I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?We don’t.
I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Solve the differential equation y′ = xey.
I The general solution of the differential equation y′ = xey is
y =− ln(
C− 12
x2)
.
I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.
I But how do we know that we did not make any mistakes?We don’t.
I So we should always check the result.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?=
x1
eln(C− 12 x2)
x1
C− 12x2
?= x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?=
x1
eln(C− 12 x2)
x1
C− 12x2
?= x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?=
x1
eln(C− 12 x2)
x1
C− 12x2
?= x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?=
x1
eln(C− 12 x2)
x1
C− 12x2
?= x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?= x
1
eln(C− 12 x2)
x1
C− 12x2
?=
x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?= x
1
eln(C− 12 x2)
x1
C− 12x2
?=
x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?= x
1
eln(C− 12 x2)
x1
C− 12x2
?= x1
C− 12x2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations
logo1
Definition An Example Double Check
Does y =− ln(
C− 12
x2)
Really Solve the
Differential Equation y′ = xey?
y′ ?= xey
ddx
[− ln
(C− 1
2x2
)]?= xe− ln(C− 1
2 x2)
− 1C− 1
2x2(−x) ?= x
1
eln(C− 12 x2)
x1
C− 12x2
= x1
C− 12x2
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separable Differential Equations