Solve the differential equation : \(x \cdot \frac{dy}{dx} - y + x \cdot \sin\left(\frac{y}{x}\right) = 0\).

Answer: \(x\tan\left(\frac{y}{2x}\right) = c\)

The equation gives \(\frac{dy}{dx} = \frac{y}{x} - \sin\left(\frac{y}{x}\right)\), which is homogeneous. Put \(y = vx\), so \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).

\(v + x\frac{dv}{dx} = v - \sin v \Rightarrow x\frac{dv}{dx} = -\sin v \Rightarrow \operatorname{cosec} v\,dv = -\frac{dx}{x}\).

Integrating: \(\log\left|\tan\frac{v}{2}\right| = -\log|x| + \log c\), so \(\log\left|x\tan\frac{v}{2}\right| = \log c\).

\(\therefore x\tan\left(\frac{y}{2x}\right) = c\) is the general solution.