# Solve : <br> `{xcos(y/x)+ysin(y/x)}ydx={ysin(y/x)-xcos(y/x)}xdy`

Updated On: 13-7-2021

3.3 k+

100+

Answer

Text Solution

The given differential equation is

{xcos(y/x)+ysin(y/x)}ydx

={ysin(y/x)-xcos(y/x)}xdy

therefore (dy)/(dx) = ({xcos(y/x)+ysin(y/x)}y)/({ysin(y/x)-xcos(y/x)}x)

=({cos(y/x)+(y/x)sin(y/x)})/(y/xsin(y/x)-cos(y/x)(y/x))

.................(1)

This is homogenous equation.

To solve it, we make the substition as

y=vx

or

(dy)/(dx)=v+x(dv)/(dx)

Substituting the values of y and

(dy)/(dx)

in equation (1), we get

v+x(dv)/(dx) = (vcosv+v^(2)sinv)/(vsinv-cosv)

or

x(dv)/(dx) = (vcosv+v^(2)sinv-v^(2)sinv+vcosv)/(vsinv-cosv)

or

x(dv)/(dx) = (2vcosv)/(vsinv-cosv)

or

int[(vsinv-cosv)/(vcosv)]dv=int(dx)/(x)

or

-log(vcosv)=2logx+logC

or

1/(vcosv)=Cx^(2)

or

-cos(y/x)=1/(Cxy)=1/C.1/(xy)

or

xycos(y/k)=k

(k=1/c)

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