If for the differential equation $y^{\mathrm{\prime }}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$

 the general solution is $y=\frac{x}{\log \left|Cx\right|}$then $\varphi (x/y)$

is given by

Putting $v=y/x$ so that $\frac{x\mathrm{d}v}{\mathrm{d}x}+v=\frac{\mathrm{d}y}{\mathrm{d}x}$we have

$\frac{x\mathrm{d}v}{\mathrm{d}x}+v=v+\varphi (1/v)\Rightarrow \frac{\mathrm{d}v}{\varphi (1/v)}=\frac{\mathrm{d}x}{x}$

$\Rightarrow \log \left|Cx\right|=\int \frac{\mathrm{d}v}{\varphi (1/v)}$ ($C$being constant of integration.)

But $y=\frac{\left|x\right|}{\log \left|\right(Cx\left)\right|}$ is the general solution so

$\begin{array}{l}\frac{x}{y}=\frac{1}{v}=\int \frac{\mathrm{d}v}{\varphi (1/v)}\Rightarrow \varphi (1/v)=-v^2\\ \Rightarrow \varphi (x/y)=-y^2/x^2.\end{array}$