Post on 08-Feb-2016
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Chapter 1: First-Order Differential Equations
1
Sec 1.4: Separable Equations and Applications
Definition 2.1
:Example
)()(
yfxg
dxdy
1
A 1st order De of the form
is said to be separable.
yx
dxdy 2
2 yxedxdy 3 yxxey
dxdy 432
3 xydxdy sin
)()( yhxgdxdy
2
How to Solve ?
)cc(c
g(x) dx h(y) dy
g(x) dxh(y) dy h(y)g(x)
dxdy
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
1 xydxdy 6 2
xedxdyxye yy 2sincos)( 2 4
xy
dxdy
1
5324
2
y
xdxdy
Sec 1.4: Separable Equations and Applications
4
3
7)0(
6
y
xydxdy
Solve the differential equation :Example2
It may or may not possible to express y in terms of x (Implicit Solution)
5324
2
y
xdxdy
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)
dxdy
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
1yx
dxdy
2
xedxdyxye yy 2sincos)( 2 4
xy
dxdy
1
5324
2
y
xdxdy
Sec 1.4: Separable Equations and Applications
9
3
7)0(
6
y
xydxdy
Solve the differential equation :Example2
42 ydxdyIt may or may not possible to express y
in terms of x (Implicit Solution)
10
11
Modeling and Separable DE
The Differential Equation
ktdtdP
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
)( TAkdtdT
Water tank with hole
ykdtdV
12
The Differential Equation
ktdtdP
K a constant
The population of 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
ktdtdP
K a constant