Solve the following differential equation: x (dy)/(dx) = y - x tan(y/x) (Application)

Put $y = vx$, so $\frac{dy}{dx} = v + x\frac{dv}{dx}$. The equation becomes $v + x\frac{dv}{dx} = v - \tan v$, so $\cot v\,dv = -\frac{dx}{x}$. Integrating: $\log|\sin v| = -\log x + \log c$, so $\sin\frac{y}{x} = \frac{c}{x}$. Solution: $x\sin\frac{y}{x} = c$