Mathematics - Differential Equation Question with Solution | TestHub

Putting $v=y/x$ so that $x\frac{dv}{dx}+v=\frac{dy}{dx}$, we have:

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

$$\Rightarrow \log |Cx|=\int \frac{dv}{\varphi (1/v)}$$

(C being the constant of integration.) But $y=\frac{x}{\log |Cx|}$ is the general solution, so:

$$\frac{x}{y}=\frac{1}{v}=\int \frac{dv}{\varphi (1/v)}\Rightarrow \varphi (1/v)=-1/v^2\Rightarrow \varphi (x/y)=-y^2/x^2$$