Solve the following differential equation: (dx)/(dy) = x/y + (f(x/y))/(f'(x/y))

The equation $\frac{dx}{dy} = \frac{x}{y} + \frac{f(x/y)}{f'(x/y)}$ is homogeneous. Put $x = vy$, so $\frac{dx}{dy} = v + y\frac{dv}{dy}$. $v + y\frac{dv}{dy} = v + \frac{f(v)}{f'(v)}$, so $\frac{f'(v)}{f(v)}dv = \frac{dy}{y}$. Integrating: $\ln|f(v)| = \ln y + \ln c$, so $f(v) = cy$. Solution: $f\left(\frac{x}{y}\right) = cy$