HOMOGENEOUS DIFFERENTIAL

EQUATIONS

Homogeneous FunctionA function f(x,y) is called Homogeneous of degree n if

Where t is a nonzero real number. Thus

are

Homogeneous function of degree 1, 8 and 0 respectively

),(),( yxftyxf n

y

xand

yx

yxxy sin..., 22

1010

Homogeneous Equation

A first order DE of the form

Is said to be Homogeneous if the function f does not depend on x and y separately, but only on ratio . Thus first order homogeneous

equation are of the form ---------(1)

A homogeneous equation

Is transformed into a separable equation (in the variables y and x) by the substitution y = vx

),( yxfdx

dy

x

y

x

yg

dx

dy

x

yg

dx

dy

Put y = vx and in eq (1)

This can be separated and be solved

)(vgdx

dvxv

dx

dvxv

dx

dy

0)( dx

dvxvgv

0)]([ xdvdxvgv

x

dx

vgv

dv

)(

02 22 dyxxyy

2

2 2

x

xyy

dx

dy

vxy Put and

Solve

Soln:

dx

dvxv

dx

dy

-----------(1)

So eq (1) becomes

vvx

xvxxv

dx

dvxv 2

2 22

22

vvdx

dvx 32

x

dx

vv

dv

3

cxvv loglog3log3

1log3

1

x

dxdv

vv 3

11

3

1

x

c

v

v 1log33

log

3

1log3

log

x

c

v

v

33

13

3 x

c

x

c

v

v

3

3 x

c

xyxy

)3(3 ycyx

0)4(52 dyyxdxyx

)4(

)52(

yx

yx

dx

dy

vxy Put and

Solve

Soln:

dx

dvxv

dx

dy

when y(1)=4

So eq(1) becomes

)4(

)52(

)4(

)52(

v

v

vxx

vxx

dx

dvxv

vv

v

dx

dvx

)4(

)52(

x

dx

vv

dvv

2)1(

)4(

-----------(1)

cxvv loglog2log2)1log(

x

dx

v

dv

v

dv

22

1

cxv

vlog

)2(

)1(log 2

2]2[]1[ x

ycx

x

y 2]2[][ xycxy

cxv

v

2)2(

)1(

2]24[]14[ c 2]24[]14[ c

12

1 c 2]2[][12 xyxy

dxyxxdxydy 22

Solve

Solve

1)1(....0)( 22 ywhendyxdxyxy

EQUATIONS REDUCIBLE TO HOMOGENEOUS

FORM

Equation Reducible to Homogeneous FormThe DE

Is not homogeneous. It can be reduced to homogeneous form as explained below

Case-I If then make the

transformation x = X + h, y = Y + k

0)()( 222111 dycybxadxcybxa

2

1

2

1

b

b

a

a

0)(

)(

22222

11111

dYckbhaYbXa

dXckbhaYbXa

Let h and k be the solution of the system of equations

Then for calculated values of h and k eq (1) will be reduced to homogeneous form

In the variables X and Y

0

0

222

111

ckbha

ckbha

0)()( 2211 dYYbXadXYbXa

Case-II If

then put

And the given equation will reduce to a separable equation in the variables x and z

ybxaz 11 2

1

2

1

b

b

a

a

Solve

Soln: Let x = X+h and y = Y+k, then

Now 5h + 5 = 0 h = -1 k = 1

32

12

yx

yx

dx

dy

3)(2

1)(2

kYhX

kYhX

dx

dy

322

122

khYX

khYX

dx

dy

032

0122

kh

kh

Put Y = vX

3212

1122

YX

YX

dx

dy

YX

YX

dx

dy

2

2

XYXY

dx

dy

21

2

v

v

dX

dvXv

21

2

vv

v

dX

dvX

21

2

cXvv lnln2)1ln(tan 21

v

v

v

vvv

dX

dvX

21

)1(2

21

22 22

X

dXdv

v

v2

1

)21(2

X

dX

v

vdv

v

dv2

1

2

1 22

cXvv lnln)1ln(tan 221

221 )1(lntan Xvcv

22

21 1lntan X

X

Yc

X

Y

)(lntan 221 YXcX

Y

])1(

)1[(ln)1(

)1(tan

2

21

y

xcx

y

Solve

Soln: Let z = 3x – 4y then

343

243

yx

yx

dx

dy

dx

dy

dx

dz43 dx

dz

dx

dy)4

1(

4

3

3

2)4

1(

4

3

z

z

dx

dz

3

2

4

3)4

1(

z

z

dx

dz

)3(4

)1()4

1(

z

z

dx

dz

)3(

)1(

z

z

dx

dz

dx

z

dzz

)1(

)3(

dxz

dzdz

)1(4

)1ln(41 zcxz

)143ln(443 1 yxcxyx

)143ln(41 yxc

yx

)143ln( yxcyx

Put z = 3x – 4y

Solve

5

1

xy

xy

dx

dy Solve

12

52

yx

yx

dx

dy