Mathematics - Differential Equation Question with Solution | TestHub

We have, $xdy=\left(y+\frac{xf(y/x)}{f^{\prime }(y/x)}\right)dx$

$\Rightarrow \frac{dy}{dx}=\frac{y}{x}+\frac{f(y/x)}{f^{\prime }(y/x)}$ which is homogenous

Putting $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$,

we obtain $v+x\frac{dv}{dx}=v+\frac{f(v)}{f^{\prime }(v)}\Rightarrow \frac{f^{\prime }(v)}{f(v)}dv=\frac{dx}{x}$

Integrating, we get $\log f(v)=\log x+\log c$

$\Rightarrow \log f(v)=\log cx\Rightarrow f\left(\frac{y}{x}\right)=cx$