Method Homogeneous Equations Reducible to separable.

Method

Homogeneous EquationsReducible to separable

Chapter 2 2

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

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Chapter 2 3

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

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Chapter 2 4

METHOD for solving Homogenous differential equations

dx

duxu

dx

dy

uxy

xduudxdy

Substitute

Substitute

OR

vyx

ydvvdydx dy

dvyv

dy

dx

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Chapter 2 5

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

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Chapter 2 6

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.

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Chapter 2 7

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

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Chapter 2 8

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)

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Chapter 2 9

.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)H

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Chapter 2 10

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)

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Chapter 2 11

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)

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Chapter 2 12

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Chapter 2 13

(1 / 3)

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Chapter 2 14

(2 / 3)

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Chapter 2 15

(3 / 3)H

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Chapter 2 16

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Chapter 2 17

(1 / 2)

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Chapter 2 18is general solution of differential equation

(2 / 2)H

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Chapter 2 19

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Chapter 2 20

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Chapter 2 21

is general solution of differential equation

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