The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{f(y/x)}{f^{\prime }(y/x)}$ is:

This is a homogeneous differential equation. Let $y=vx⟹\frac{dy}{dx}=v+x\frac{dv}{dx}$.Substituting into the equation:$v+x\frac{dv}{dx}=v+\frac{f(v)}{f^{\prime }(v)}$$x\frac{dv}{dx}=\frac{f(v)}{f^{\prime }(v)}$ Separating variables:$\frac{f^{\prime }(v)}{f(v)}dv=\frac{dx}{x}$ Integrating both sides:$\int \frac{f^{\prime }(v)}{f(v)}dv=\int \frac{1}{x}dx$$\ln |f(v)|=\ln |x|+\ln C⟹f(v)=Cx$ Replacing $v=y/x$:$f(y/x)=Cx$.