Mathematics - Differential Equation Question with Solution | TestHub

Putting $v=y/x$ so that $\frac{xdv}{dx}+v=\frac{dy}{dx}$,

we have $\frac{xdv}{dx}+v=v+\varphi (1/v)\Rightarrow \frac{dv}{\varphi (1/v)}=\frac{dx}{x}$

$\Rightarrow \log |Cx|=\int \frac{dv}{\varphi (1/v)}$ (being constant of integration.)

But $y=\frac{x}{\log |Cx|}$ is the general solution so $\frac{x}{y}=\frac{1}{v}=\int \frac{dv}{\varphi (1/v)}\Rightarrow \varphi (1/v)=-1/v^2\Rightarrow \varphi (x/y)=-y^2/x^2$.