Post on 30-Dec-2015
Chapter 1: First-Order Differential Equations
1
Sec 1.4: Separable Equations and Applications
Definition 2.1
:Example
)(
)(
yf
xg
dx
dy
1
A 1st order De of the form
is said to be separable.
y
x
dx
dy 2
2 yxedx
dy 3 yxxeydx
dy 432
3 xydx
dysin
)()( yhxgdx
dy
2
How to Solve ?
)cc(c
g(x) dx h(y) dy
g(x) dxh(y) dy
h(y)
g(x)
dx
dy
21constant oneEnough 2)
nintegratio ofConstant FORGET DONOT 1) :Note
sidesboth integrate :Step3
rewrite:Step2
Separable ifchek :Step1
:Solution of Method
Sec 1.2
3
:Example
1y
x
dx
dy2
xedx
dyxye yy 2sincos)( 2 4
x
y
dx
dy
1
53
242
y
x
dx
dy
Sec 1.4: Separable Equations and Applications
4
3
7)0(
6
y
xydx
dy
Solve the differential equation :Example2
It may or may not possible to express y in terms of x (Implicit Solution)
53
242
y
x
dx
dy
Sec 1.4: Separable Equations and Applications
5
Solve the IVP :Example2
3)1( y
0' yyx
Implicit Solutions and Singular Solutions
6
Solve the IVP :Example2
2)0( y
:Example2 Implicit So , Particular, sol
2
2
-2
-2
How to Solve ?
)cc(c
g(x) dx h(y) dy
g(x) dxh(y) dy
h(y)
g(x)
dx
dy
21constant oneEnough 2)
nintegratio ofConstant FORGET DONOT 1) :Note
sidesboth integrate :Step3
rewrite:Step2
Separable ifchek :Step1
:Solution of Method
Sec 1.2
7
Remember division
3) Remember division
3/2)1(6' yxy
Implicit Solutions and Singular Solutions
8
Solve the IVP :Example2 :Example2Singular Soldivision
:Remark
a general Sol
Particular Sol
Family of sol (c1,c2,..)
No C
:Remark
a general Sol
The general Sol
Family of sol (c1,c2,..)
1) It is a general sol2) Contains every
particular sol
:Remark
Singular Sol no value of C gives this sol
:Example
1y
x
dx
dy2
xedx
dyxye yy 2sincos)( 2 4
x
y
dx
dy
1
53
242
y
x
dx
dy
Sec 1.4: Separable Equations and Applications
9
3
7)0(
6
y
xydx
dy
Solve the differential equation :Example2
42 ydx
dyIt may or may not possible to express y in terms of x (Implicit Solution)
10
11
Modeling and Separable DE
The Differential Equation
ktdt
dP K a constant
serves as a mathematical model for a remarkably wide range of natural phenomena.
Population GrowthCompound InterestRadioactive DecayDrug Elimination
According to Newton’s Law of cooling
)( TAkdt
dT
Natural Growth and Decay Cooling and Heating
Water tank with hole
ykdt
dV
Torricelli’s Law
12
The Differential Equation
ktdt
dP K a constant
The population f a town grows at a rate proportional to the population present at time t. the initial population of 500 increases by 15% in 10 years. What will be the population in 40 years?
13
The Differential Equation
ktdt
dP K a constant