(x sin y/x - y cos y/x) dx + x cos y/x dy = 0

∴ dy/dx \(=\frac{y\,cos\,y/x\,-\,x\,sin\,y/x}{x\,cos\,y/x}\)

⇒ dy/dx \(=\frac{y/x\,cos\,y/x\,-\,sin\,y/x}{cos\,y/x}\)  ...(1)

Let y = vx

∴ dy/dx = v + x dv/dx

∴ From (1), v + x dv/dx \(=\frac{v\,cos\,v-sin\,v}{cos\,v}\)

⇒ x dv/dx \(=\frac{v\,cos\,v-sin\,v}{cos\,v}-v\)

\(=\frac{v\,cos\,v-sin\,v-v\,cos\,v}{cos\,v}\)

= -sin v/cos v = -tan v

⇒ cot v dv = -1/x dx

⇒ ∫ cos v/sin v dv = -∫1/x dx

⇒ log sin v = -log x + log c

⇒ log sin v = log c/x

⇒ sin v = c/x

⇒ x sin v = c

⇒ x sin y/x = c  ...(2) (By putting v = y/x)

Equation (2) represent the solution of given differential equation.