The general solution of $y^{\mathrm{\prime }}=\frac{y}{x}+\frac{x}{y}\frac{\varphi \left(y^2/x^2\right)}{{\varphi }^{\mathrm{\prime }}\left(y^2/x^2\right)}$ is 

Putting  $\frac{y}{x}=u,$we have $\frac{dy}{dx}=u+x\frac{du}{dx}$.

given differential equation can be written as 

$\begin{array}{l}\mathrm{u} + \mathrm{x}\frac{du}{dx}=u+\frac{1}{u}\frac{\varphi \left(u^2\right)}{\varphi \left(u^2\right)}\\ \Rightarrow \frac{u{\varphi }^{\mathrm{\prime }}\left(u^2\right)}{\varphi \left(u^2\right)}du=\frac{dx}{x}\\ \Rightarrow \log \varphi \left(u^2\right)=\log x^2+\text{ const }\Rightarrow x^2=k\varphi \left(\frac{y^2}{x^2}\right)\end{array}$