The general solution of the differential equation $(x\sin \frac{y}{x})dy=(y\sin \frac{y}{x}-x)dx$ is 

$\Rightarrow x\sin \left(\frac{y}{x}\right)\frac{dy}{dx}=y\sin \left(\frac{y}{x}\right)-x$

Put y=vx

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{v}+\mathrm{x}\frac{\mathrm{dv}}{\mathrm{dx}}$

$\begin{array}{r}\therefore \mathrm{x}\left(\mathrm{x}\frac{\mathrm{dv}}{\mathrm{dx}}+\mathrm{v}\right)\sin \mathrm{v}=-\mathrm{x}+\mathrm{xvsin}\mathrm{v}\\ \Rightarrow \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{-\csc \mathrm{v}}{\mathrm{x}}\Rightarrow \sin \mathrm{v}\frac{\mathrm{dv}}{\mathrm{dx}}=-\frac{1}{\mathrm{x}}\end{array}$

Integrating on both sides

$\begin{array}{r}\int \sin v\frac{dv}{dx}dx=-\int \frac{1}{x}dx\\ \Rightarrow -\cos \mathrm{v}=-\log \mathrm{x}+\mathrm{k}\\ \Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\log \mathrm{x}+\mathrm{k}\end{array}$