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