The general solution of the differential equation.
\(
\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right] d y=0 \text { is }
\)

We have \(\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)~+\) \(\cos \left(\frac{y}{x}\right)] d y=0\)
\(
\therefore \frac{d y}{d x}=\frac{\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right)}{\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)}
\)
Put \(\frac{y}{x}=v \Rightarrow y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\( \therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}}{\frac{1}{\mathrm{v}} \sin \mathrm{v}+\cos \mathrm{v}}=\frac{\mathrm{v}^2 \cos \mathrm{v}}{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}} \Rightarrow \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}=\) \(\frac{-\mathrm{v} \sin \mathrm{v}}{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}} \)
\( \therefore \int \frac{\sin \mathrm{v}+\mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}} \mathrm{dv}=\int \frac{-\mathrm{dx}}{\mathrm{x}} \Rightarrow \int \frac{1}{\mathrm{v}} \mathrm{dv}~+\) \(\int \cot \mathrm{v} d \mathrm{v}=-\int \frac{\mathrm{dx}}{\mathrm{x}} \)
\( \therefore \log |\mathrm{v}|+\log |+\sin \mathrm{x}|=-\log |\mathrm{x}|~+\) \(\log \mathrm{k} \Rightarrow \log |(\mathrm{v})(\sin \mathrm{v})(\mathrm{x})| \)
\( =\log \mathrm{k}\)
\(\therefore y \sin \left(\frac{y}{x}\right)=k\)