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