Chapter 2 1

Chapter 2 2

Chapter 2 3

Example

Chapter 2 4

Chapter 2 5

EXAMPLE

Chapter 2 6

Solution

Chapter 2 7

Chapter 2 8

Chapter 2 9

Method for solving First Order

Differential Equations

Methods

Variable SeparableVariable Separable

Reducible to variable separable Reducible to variable separable

Exact Differential EquationExact Differential Equation

Integrating FactorIntegrating Factor

Chapter 2

Separable Variable

x is independent variable and y is dependent variable

or

are separable forms of the differential equation

or

General solution can be solved by directly integrating both the sides

+ cWhere c is constant of integration

11DO YOU REMEMBER INTEGRATION FORMULA

Separation of Variables

and are separable

but is not separable.

xy xy y

y

x yy

x y

Definition A differential equation of the type y’ = f(x)g(y) is separable.

Example

x

yy

Example

Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly.

2 2

2 212 2

y xC y x C

ydy xdx

Chapter 2 13

EXAMPLE:

Chapter 2 14

EXAMPLE:

Chapter 2 15

To find the particular solution, we apply the given initial condition, when x =1, y = 3

is solution of initial value problem

Chapter 2 16

Chapter 2 17

Chapter 2 18

xdxdyyy 2ln2

xdxdyyy 2ln2

cxyyyy 22 ln

xdxdyyey y 22

xdxdyyey y 22

cxeyey yy 22

211 1sinsin yyyydy

xdxdyyy

xdxdyyy

2sin2

2sin21

1

cxyyyy 2212 1sin

Note1: If we have

Integrating by parts

Note.2. If we have

Integrating by parts

Note.3. If we have

yyyydy lnln

yyy eeydyye

Chapter 2 19

Chapter 2 20

Chapter 2 21

Method

Homogeneous EquationsReducible to separable

Chapter 2 23

Homogenous Differential Equations

A differential equation

Homogenous differential equation if

every t, where t R

isyxfdx

dy ),(

, , nf tx ty t f x y for

Chapter 2 24

Example:1. Show that differential equation is homogenous differential equation.

dxyxxydy 22

Solution: xy

yx

dx

dy 22

xy

yxyxf

22

,

xyt

ytxttytxf

2

2222

,

2 2 2 2 2

2,

t x y x yf x y

t xy xy

Differential equation is homogeneous

Differential equation is homogeneous

Chapter 2 25

METHOD for solving Homogenous differential equations

dx

duxu

dx

dy

uxy

xduudxdy

Substitute

Substitute

OR

vyx

ydvvdydx dy

dvyv

dy

dx

Chapter 2 26

Using substitution the homogeneous differential equation

is reduce to separable variable form.Example:2 Solve the homogenous differential

equation

xy

yx

dx

dy 22

Solution:Rewriting in the form : 0,, dyyxNdxyxM

.

022 xydydxyxuxy xduudxdy substitute and

Chapter 2 27

02222 xduudxuxdxuxx0322222 duuxdxxudxuxdxx

032 duuxdxx

duuxdxx 32

udux

dxx

3

2

udux

dx is variable separable form

udxx

dx cu

x 2

ln2

cx

yx

2

2

2

1ln is general solution.

Chapter 2 28

Note. Selection of substitution Differential Equation depends on

number of terms of coefficients yxandyxM ,N ,

01321 dydx uxy 1.

If , then take

2.

If

03211 dydx , then take x vx

3.

If 02121 dydx , then take

x = vy or y = ux

Chapter 2 29

Example:. Solve the Differential Equation by using appropriate substitution

0222 dyxdxxxyy Solution: Differential equation is homogeneous as degree of each term is same, hence we can use either y = ux or x = vy as substitution

xduudxdy

uxy

Let

Substituting y and dy in the given equation

duxudxxdxxdxuxdxxu

xduudxxdxxuxxu322222

22222

duxdxux

duxdxxdxxu322

3222

1

(1 / 2)

Chapter 2 30

.tanln

tanln

1

1

1

2

cx

yx

cux

u

du

x

dx

21 u

du

x

dx

is Separable form

Integrating both the sides

is general solution of the differential equation

Separating variable u and x (2 / 2)

Chapter 2 31

Example: Show that differential equation

22 943 yxdx

dyxy is homogeneous

dxyxxydy 22 943Solution:

xdvudxdyuxy ,

2 2 2

2 2 3 2 2 2

3 . 4 9

3 3 4 9

x ux udx xdu x u x dx

x u dx ux du x dx u x dx

dxuxdxxudxxduux 222223 64643

(1 / 2)

Chapter 2 32

x

dx

u

udu

264

3

x

dx

u

udu264

3

ududz

uz

12

64 2

Let

x

dx

z

dz

4

1

.ln64l4

1

lnln4

1

2

2

cxx

yn

cxz

is general solution of the differential equation

is Separable form

Integrating both the sides

(2 / 2)

Chapter 2 33

Hom

ogen

eous

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 34

(1 / 3)

Hom

ogen

eous

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 35

(2 / 3)

Hom

ogen

eous

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 36

(3 / 3)H

omog

eneo

us D

iffer

entia

l Equ

ation

C

hapt

er 2

Chapter 2 37

Hom

ogen

eous

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 38

(1 / 2)

Hom

ogen

eous

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 39is general solution of differential equation

(2 / 2)H

omog

eneo

us D

iffer

entia

l Equ

ation

C

hapt

er 2

Chapter 2 40

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 41

Diff

eren

tial E

quati

on

Cha

pter

2

Chapter 2 42

is general solution of differential equation

Diff

eren

tial E

quati

on

Cha

pter

2