Question

The solution of the differential equation $xdy=[y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)}]dx$ is

• A. $f\left(\frac{y}{x}\right)=cx$
• B. $f\left(\frac{y}{x}\right)=cy$
• C. $f\left(\frac{x}{y}\right)=cx$
• D. $f\left(\frac{x}{y}\right)=c{x}^2$

Answer 1

The correct option is A $f\left(\frac{y}{x}\right)=cx$

$xdy=[y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)}]dx$
$\Rightarrow \frac{dy}{dx}=\frac{y}{x}+\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)}$
$\Rightarrow v+x\frac{dv}{dx}=v+\frac{f\left(v\right)}{f^{\prime }\left(v\right)}$ where $v=\frac{y}{x}$
$\Rightarrow \frac{f^{\prime }\left(v\right)}{f\left(v\right)}dv=\frac{dx}{x}$
$\Rightarrow f\left(v\right)=xc$ (where $v=\frac{y}{x}$
Hence (a) is correct answer.