The solution of the differential equation $\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi'\left(\frac{y}{x}\right)}$, is

$\phi\left(\frac{y}{x}\right)=k x$

Substituting $y=v x$ and $\frac{d y}{d x}=v+x \frac{d v}{d x}$, we get

$v+x \frac{d v}{d x}=v+\frac{\phi(v)}{\phi'(v)} \Rightarrow \frac{\phi'(v)}{\phi(v)} d v=\frac{1}{x} d x$

On integrating, we get

$\log \phi(v)=\log x+\log k \Rightarrow \phi(v)=k x \Rightarrow \phi\left(\frac{y}{x}\right)=k x$