Solve the following differential equation:

Given the differential equation $\frac{dx}{dy} = \frac{x}{y} + \frac{f(x/y)}{f'(x/y)}$. This is a homogeneous differential equation of the form $\frac{dx}{dy} = F(x/y)$. Substitute $x = vy \Rightarrow \frac{dx}{dy} = v + y\frac{dv}{dy}$. Then: $v + y\frac{dv}{dy} = v + \frac{f(v)}{f'(v)} \Rightarrow y\frac{dv}{dy} = \frac{f(v)}{f'(v)}$. Separating variables: $\frac{f'(v)}{f(v)} dv = \frac{dy}{y}$. Integrating both sides: $\int \frac{f'(v)}{f(v)} dv = \int \frac{dy}{y} + \ln c \Rightarrow \ln|f(v)| = \ln y + \ln c = \ln(cy)$. Exponentiating gives $f(v) = cy$. Substituting back $v = \frac{x}{y}$: $f\left(\frac{x}{y}\right) = cy$, which is the required general solution.