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

Take $\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{V}$

diff w.r.to x on both sides

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx} \\ Given equatiion becomes \\ v+x\frac{dv}{dx}=v+\frac{\varphi \left(v\right)}{\varphi \text{'}\left(v\right)} \\ \Rightarrow \frac{\varphi \text{'}\left(v\right)}{\varphi \left(v\right)}dv=\frac{dx}{x} \\ Integrating on both sides \\ \int \frac{\varphi \text{'}\left(v\right)}{\varphi \left(v\right)}dv=\int \frac{dx}{x} \\ \Rightarrow \log \left|\varphi \left(v\right)\right|=\log \left|x\right|+\log k \\ \Rightarrow \varphi \left(v\right)=kx \\ \Rightarrow \varphi \left(\frac{y}{x}\right)=kx \end{gathered}$$