The solution of the differential equation \(x \cdot \sin \left(\frac{y}{x}\right) d y=\left[y \cdot \sin \left(\frac{y}{x}\right)-x\right] d x\) is

We have \(x \cdot \sin \left(\frac{y}{x}\right) d y=\left[y \cdot \sin \left(\frac{y}{x}\right)-x\right] d x\)
\(\therefore \frac{d y}{d x}=\frac{y \cdot \sin \left(\frac{y}{x}\right)-x}{x \sin \left(\frac{y}{x}\right)}=\left(\frac{y}{x}\right)-\frac{1}{\sin \left(\frac{y}{x}\right)}\)
\(\therefore \frac{d y}{d x}-\left(\frac{1}{x}\right) y=-\operatorname{cosec}\left(\frac{y}{x}\right)\)
Put \(\frac{y}{x}=v \Rightarrow y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore\left(\mathrm{v}+\frac{\mathrm{xdv}}{\mathrm{dx}}\right)-\mathrm{v}=-\operatorname{cosec} \mathrm{v} \Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=-\operatorname{cosec} \mathrm{v}\)
\(\therefore \int \frac{\mathrm{dv}}{\operatorname{cosec} \mathrm{v}}=-\int \frac{\mathrm{d} \mathrm{x}}{\mathrm{x}} \Rightarrow \int \sin \mathrm{v} \mathrm{dv}=-\log |\mathrm{x}|\)
\(-\cos v=-\log |x|+c_{1} \Rightarrow \log x+c=\cos v\)
\(\therefore \cos \left(\frac{y}{x}\right)=\log |x|+c\)