Solve the following differential equation: `x(dy)/(dx)=y-xcos^2(y/x)`

Explanation

`x(dy)/(dx)=y-xcos^(2)"(y)/(x)`
`implies (dy)/(dx)=(y)/(x)-cos^(2)"(y)/(x)`………..`(1)`
It is a homogenous differential equation.
Let `y=vx`
`implies (dy)/(dx)=v+x*(dv)/(dx)`
Put this value in eq. `(1)`
`v+x(dv)/(dx)=v-cos^(2)v`
`impliesx(dv)/(dx)=-cos^(2)v`
`implies sec^(2)vdv=-(dx)/(x)`
`implies intsec^(2)vdv=-int(dx)/(x)`
`implies tanv=-logx+c`
`implies tan"(y)/(x)+logx=c`.