Show that the given differential equation is homogeneous and solve each of them.
`{xcos((y)/(x))+ysin((y)/(x))}ydx={ysin((y)/(x))-xcos((y)/(x))}xdy`

Explanation

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)`