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

1yx

dxdy

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

Natural Growth and Decay Cooling and Heating

Water tank with hole

ykdtdV

Torricelli’s Law

12

The Differential Equation

ktdtdP

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

ktdtdP

K a constant