3.homogeneous differential equations

Homogeneous Differential Equation:

A differential equation of the form

y,x

y,xf

dx

dy

,

is called a homogeneous differential equation if y,xf and y,x are

homogeneous functions of the same degree in x and y.

Now, since y,xf and y,x are homogeneous functions of the same

degree in x and y, then a homogeneous differential equation can also be

written as

x

yg

dx

dy.

Method to solve a homogeneous differential equation :

Given differential equation is

x

yg

dx

dy.

3rd TopicDifferential Equations of First order

“Homogeneous Equations”

(Reduction to a Separable Form)

Prepared by:

Dr. Sunil

NIT Hamirpur (HP)

(Last updated on 01-02-2011)

Differential Equations of First Order: Homogeneous Equations

Prepared by: Dr. Sunil, NIT Hamirpur

2

(i) Put vxy , and then dx

dvxv

dx

dy in the given differential equation, we get

)v(gdx

dvxv , which reduces to a separable equation.

(ii) Separating the variables v and x, we get x

dx

v)v(g

dv

.

(iii) Integrating on both sides, we get the required solution.

cx

dx

v)v(g

dv

(iv) At last, replace v by x

y, in the solution obtained in (iii).

Now let us solve some homogeneous differential equations

Homogeneous differential equations: 19 (solved)

Home assignments: 20 problems

Q.No.1.: Solve 0xydydxyx 22 .

Sol.: Given equation is

x

yx

y1

xy

yx

dx

dy 2

2

22

, (i)

which is homogeneous differential equation in x and y.

Putting vxy , then dx

dvxv

dx

dy .

Equation (i) becomes v

v1

dx

dvxv

2

v

v21v

v

v1

dx

dvx

22

.

Separating the variables, we get x

dxdv

v21

v2

.

Integrating both sides, we get

cx

dxdv

v21

v2

c

x

dxdv

v21

v4

4

12

cxlogv21log

4

1 2

c4v21logxlog4 2 c4v21xlog 24



3

cex

y21x c3

2

24

(where c4'c )

cy2xx 222 .

This is the required solution of this homogeneous differential equation.

Q.No.2.: Solve 0dyy

x1edxe1 y/xy/x

.

Sol.: The given equation can be written as y/x

y/x

e1

y

x1e

dy

dx

, (i)


Putting vyx , then dy

dvyv

dy

dx .

Equation (i) becomes

v

v

e1

v1e

dy

dvyv

v

v

v

vv

v

v

e1

ev

e1

e1vv1ev

e1

v1e

dy

dvy

.

Separating the variables, we get

v

v

v

v

ev

evddv

ev

e1

y

dy

.


cev

evd

y

dyv

v

cevlogylog v cv eevy cyex y/x [say].


Q.No.3.: Solve the differential equation 22 xy3dx

dy.xy2 .

Sol.: Here the given equation is 22 xy3dx

dy.xy2

xy2

xy3

dx

dy 22 , (i)



dvxv

dx

dy .



4

Equation (i) becomes v2

1v3

vx2

xxv3

dx

dv.xv

2

2

222

v2

1v

v2

v21v3

dx

dvx

222

.


dx

v1

vdv22


clogx

dx

v1

vdv22

clogxlogv1log 2 clog

x

v1log

2

cx

yx3

22

322 cxyx .


Q.No.4.: Solve the differential equation 0xydydxy2x 22 .

Sol.: The given equation is xy

y2x

dx

dy 2 , (i)


Put vxy , then dx

dvxv

dx

dy .

Equation (i) becomes v

vv21v

v

v21

dx

dvx

vx

xv2x

dx

dvxv

222

2

222

.


dxdv

v1

v2

.


clogxlogv1log2

1clog

x

dxdv.

v1

v2

2

1 22

122222 cxv1clogxlogv1log 1

22

cxx

y1

422 cxyx .




5

Q.No.5.: Solve the differential equation dxyxydxxdy 22 .

Sol.: The given equation is dxyxydxxdy 22

x

yyx

dx

dy 22

. (i)

Since given equation is homogeneous differential equation in x and y.


dvxv

dx

dy .


x

yyx

dx

dy 22 vv1

dx

dv.xv 2 2v1

dx

dv.x .


dx

v1

dv2

.


clogxlogv1vlogclogx

dx

v1

dv 2

2

xcv1v 2 .

Putting x

yv , we get 2222

2

2

cxxyxyxcx

y1

x

y .


Q.No.6.: Solve the differential equation xydy2dxyx 22 .

Sol.: The given equation is xydy2dxyx 22

xy2

yx

dx

dy 22

. (i)

which is a homogeneous differential equation in x and y.

Putting vxy , so that dx

dvxv

dx

dy .



6


v2

v1

vx2

vxx

dx

dvxt

2

2

222

v2

v31v

v2

v1

dx

dvx

22

.


dx

v31

vdv22

.

Integrating both sides, we get clogx

dx

v31

vdv22

(ii)

Putting 2v31y dy6

1vdvvdv6dy , we get

xclogclogxlogy

dy

3

1 xclogylog

3

1 xcy 3/1 .

Putting 2t31y in above equation, we get xct313/12

.

Putting

x

yt in above equation, we get

xcx

y31

3/1

2

2

cxx

y3x 31

2

22

cxy3x

x 322

2

c

1y3xx

122

cy3xx 22 .


Q.No.7.: Solve the differential equation dyxy2xdxxy2y 22

Sol.: Here the given equation is xy2x

xy2y

dx

dy2

2

, (i)


Put vxy , then dx

dvxv

dx

dy .

Equation (i) becomes v21

v2v

xy2x

vx2xv

dx

dvxv

2

2

222

v

v21

v2v

dx

dvx

2

v21

v2vv2v

dx

dvx

22

.



7


dxdv

vv3

v212

.


clogx

dxdv

vv

1v2

3

12

clogxlogvvlog

3

1 2

332 clogxlogvvlog 332 clogxvvlog 332

2

cxx

y

x

y

32

32

cx

xyxy

12

23

cx

yxyx

cxyxy .


Q.No.8.: Solve the differential equation 0dyyxydxx 332 .

Sol.: The given equation is 0dyyxydxx 332

33

2

yx

yx

dx

dy

, (i)


Putting vxy , so that dx

dvxv

dx

dy .

Equation (i) becomes 333

3

v1

v

v1x

vx

dx

dvxv

3

4

3 v1

vv

v1

v

dx

dvx

.


dxdy

v

v14

3

.


dxdv

v

1

v

14

clogx

dxdv

v

1dvv 4 clogxlogvlog

3

v 3

clogvxlog3

v 3

3v

1

3

1ylogclog

3

3

y

x

3

1cylog



8

cylog3y

x3

.


Q.No.9.: Solve the differential equation dxyxxdyydx 22 .

Sol.: The given equation is dxyxyxdy 22

x

yxy

dx

dy 22 . (i)

Since the given equation is homogeneous differential equation in x and y.

Putting dx

dvv

dx

dy vxy .

Equation (i) becomes 2v1vdx

dv.xv 2v1

dx

dv.x .


dx

v1

dv2

.


clogxlogv1vlogcx

dx

v1

dv 22

x

clogv1vlog 2

x

cv1v 2 .

Putting x

yv , we get

cyxyx

c

x

yx

x

y 2222

.


Q.No.10.: Solve the differential equation dx

dyxy

dx

dyxy 22 .

Sol.: The given equation is dx

dyxy

dx

dyxy 22



9

dx

dy.1

x

yxy 22

1x

yx

y

dx

dy 2

2

. (i)


Put vxy dx

dvxv

dx

dy .

Equation (i) becomes 1v

v

dx

dvxv

2

1v

vvv

dx

dvx

22

.


dxdv

v

1v

.


dxdv

v

1v

clogx

dxdv

v

11

clogxlogvlogv

By putting x

yv , we get

xclogx

ylog

x

y .


Q.No.11.: Solve the differential equation xdyydxxy3dyydxx 33

Sol.: The given equation is ydyx3dxxy3dyydxx 2233

dy.yx3ydxxy3x 2323

x

y3

x

y

x

y31

yx3y

xy3x

dx

dy3

2

23

23

. (i)


Put vxy dx

dvxv

dx

dy .

Equation (i) becomes v3v

v3vv31

dx

dvx

v3v

v31

dx

dvxv

3

242

3

2



10

x

dxdv

v1

v3v4

3

.


dxdv

v1

v3v4

3

clogxlogdvv1

v3dv

v1

v44

3

xclogdvv1

v2

2

3dv

v1

v4

4

144

3

Put duvdv2uv2 , we get

xclogduu1

1

2

3dv

v1

v4

4

124

3

xclogu1

u1log

4

3v1log

4

1 4

c

xa

xalog

a2

1

xa

dx22

xclogv1

v1log

4

1v1log

4

13

2

24

xclogv1

v1log

4

122

42

xclogv1

v1log

2/12

12

xc

v1

v12

2

22222 v1cxv1

2222

2

x

y1cx

x

y1

222222 yxcyx

22222 xy'cyx .


Q.No.12.: Solve the differential equation x

ysin

x

y

dx

dy

Sol.: The given equation is x

ysin

x

y

dx

dy . (i)


Put vxy dx

dvxv

dx

dy .



11

Equation (i) becomes x

dx

xsin

dvvsinv

dx

dvxv .


dxecvdvcos

clogxlogcotv- veccoslog xclogvsin

vcos1log

xclog

2

vcos

2

vsin2

x

vsin2

log

2

cxtanx2yxcx2

ytanxc

2

vtan 1 .


Q.No.13.: Solve the differential equation 0dyx

ysecxdx

x

ysecy

x

ytanx 22

.

Sol.: The given equation is 0dyx

ysecxdx

x

ysecy

x

ytanx 22

x

ysec

x

ytan

x

y

x

ysecx

x

ytanx

x

ysecy

dx

dy

22

2

. (i)


Put vxy dx

dvxv

dx

dy .

Equation (i) becomes vsec

vtanv

dx

dvxv

2

x

dxdv

vtan

vsec2

.


dxdv

vtan

vsec2

clogxlogvtanlog clogvtanxlog

cvtanx cx

ytanx .


Q.No.14.: Solve the differential equation dyyxedxye 2y/xy/x



12

Sol.: The given equation is dyyxedxye 2y/xy/x

y/x

2y/x

ye

yxe

dy

dx

y/xx

y

y

x

dy

dx , (i)


Putting vyx dy

dvyv

dy

dx .

Equation (i) becomes ve

yv

dy

dvyv 0e

dy

dvy v

ve

dy

dv .

Separating the variable, we get dye

dvv .

Integrating both sides, we get cdye

dvv

cyev cye y/x .


Q.No.15.: Solve the differential equation 0dyy

xlogxydx

y

xlogxy 22

Sol.: The given equation is 0dyy

xlogxydx

y

xlogxy 22

y

xlog

y

x

1y

xlog

y

x

dy

dx

2

, (i)


Putting vyx dy

dvyv

dy

dx .


vlogv

1vlogv

dy

dvyv

2

vlogv

vlogv1vlogv

dy

dvy

22

vlogv

1

dy

dvy

.



13

Separating the variable, we get vdvlogvy

dy

Integrating both sides, we get cvdvlogvy

dy

cdvv

1.

2

v

2

v.vlogylog

22

cvdv

2

1

2

v.vlogylog

2

c4

v

2

v.vlogylog

22

c

y4

x

y

xlog

y2

xylog

2

2

2

2

cy4

x

x

ylog

y2

xylog

2

21

2

2

cy4

x

x

ylog

y2

xylog

2

2

2

2

c1x

ylog2

y4

xylog

2

2

.


Q.No.16.: Solve the differential equation 0xdyydxx

ysinxdx 2 .

Sol.: The given equation is 0xdyydxx

ysinxdx 2

0dx

dyxy

x

ysinx 2

x

y

x

ysin

1

dx

dy

2 .


Putting vxy dx

dvxv

dx

dy .

Equation (i) becomes vvsin

1

dx

dvxv

2

x

dxvdvsin2 .


'cxlogdv2

v2cos1'c

x

dxvdvsin2

'cxlog2

x

ysin

x

y

2

12

'cxlog2

vsinv

2

1 2



14

cx

y2sin

2

1

x

y

2

1xlog

, where 'cc .


Q.No.17.: Solve the differential equation 0dyyx2dxy2x .

Sol.: Given equation is 0dyyx2dxy2x

0

yx2

y2x

dx

dy

, (i)


Put x

yu , then

dx

dyu

dx

xdu .

Equation (i) becomes u2

u21

dx

xduu

.

2u

1u4uu

u2

u21

dx

xdu 2

.

Separating the variables, we get

1u4u

1u4ud

2

1

1u4u

du2u

x

dx2

2

2

.


02 c1u4ulog xlog2 c1u4ux 22 c1

x

y4

x

yx

2

22

cxxy4y 22 .


Q.No.18.: Solve the differential equation 0yy

x1.e2e21 y/xy/x

.

Sol.: Given equation is 0yy

x1.e2e21 y/xy/x

0dyy

x1e2dxe21 y/xy/x

, (i)


Put uy

x , so that udyydudx .



15

Equation (i) becomes 0dyu1e2yduudye21 uu

0due21ydye2u uu .

Separating the variables, we get 0due2u

e21

y

dyu

u

.


clog2eulogylog u ce2uy u .

Replacing u, we get

ce2y

xy y/x

.


Q.No.19.: Solve the differential equation 0xdydxyxy 22

, y(1) = 0.

Sol.: Given equation is 0xdydxyxy 22

222

x

y1

x

y

x

yxy

dx

dy

,


Put y = ux, then dx

duxu

dx

dy .

Equation (i) becomes 2u1udx

duxu 2u1

dx

dux

Separating the variables, we get 1u

du

x

dx2

.


1uulogclogxlog 2 cx1uu 2

Replacing u, we have

cx1x

y

x

y2

2

222 cxyxy .

Put y = 0, when x = 1, then c = 1.



16

So, the required solution is

222 xyxy .

Home AssignmentsQ.No.1.: Solve the differential equation 0dyyx3xdxxy2yx 2322 .

Ans.: y/x23 excy .

Q.No.2.: Solve the differential equation 0dyxydxyx .

Ans.: cx

ytan2yxlog 122 .

Q.No.3.: Solve the differential equation yx

y

dx

dyx

2

.

Ans.: y/xecx .

Q.No.4.: Solve the differential equation yxydx

dyyxx .

Ans.: cy

xlog

y

x .

Q.No.5.: Solve the differential equation 0ydxdyxxy .

Ans.: cylogy

x2 .

Q.No.6.: Solve the differential equation 0dyxy3x2dxxy2y 22 .

Ans.: cyxxy2 .

Q.No.7.: Solve the differential equation xlogylogydx

dyx .

Ans.: cx1xey .

Q.No.8.: Solve the differential equationx

ycosxy

dx

dyx 2 .

Ans.: cxlogx

ytan .



17

Q.No.9.: Solve the differential equation

0dx

dyx

x

ycosx

x

ysinyy

x

ysiny

x

ycosx

.

Ans.: cx

ycosxy .

Q.No.10.: Solve the differential equation 0dyxxy2dxy3xy2 22 .

Ans.: 32 cxxyy .

Q.No.11.: Solve the differential equation 0dyyxxydxyxyx 222223

.

Ans.: 332/322 cxlogxyx .

Q.No.12.: Solve the differential equation 0dyyx4dxy5x2 , y(1) = 4.

Ans.: xy12yx2 2 .

Q.No.13.: Solve the differential equation xx

ysiny

dx

dy

x

ysin.x .

Ans.: 0cxlogx

ycos .

Q.No.14.: Solve the differential equation xyx

ytanyx3yx 1222 .

Ans.: 3cxtanxy .

Q.No.15.: Solve the differential equation dxyxdyxyx 222 .

Ans.: x/y2 e.cxyx .

Q.No.16.: Solve the differential equation 0dxyxy2yxxydyx 32232

Ans.: yx

x2

xyx

xyclog

4

.


x

ycos.xyyx 2 ,

41y

.

Ans.: x

ytan xlog1 .



18

Q.No.19.: Solve the differential equation 22

22

yxy8x6

y2xy5x6y

.

Ans.: 129 x2ycx3yxy .


0dyx

ysecx

x

ycos.xdx

x

ysecy

x

ycosy

x

ytanx2

x

ysinx2 22

Ans.: cx

ytan

x

ysinx 2

.

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3.homogeneous differential equations

Documents

given differential equation

given equation

separable equation

homogeneous functions

required solution

form y

variables v

rd topicdifferential