Solution of the differential equation  xsinyxdy=ysinyx−xdx is

We have dydx=ysin⁡yx−xxsin⁡yx put y=Vx So that dydx=V+xdVdx Hence, V+xdVdx=Vsin⁡V−1sin⁡V=V−1sin⁡V⇒xdVdx=−1sin⁡V⇒∫dxx+∫sin⁡VdV=c⇒log⁡x−cos⁡V=c⇒log⁡x=cos⁡yx+c