Question

Solution of the equation $xdy=(y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$

• A. $f\left(\frac{x}{y}\right)=cy$
• B. $f\left(\frac{y}{x}\right)=cx$
• C. $f\left(\frac{y}{x}\right)=cxy$
• D. $f\left(\frac{y}{x}\right)=0$

Answer 1

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

We have, $xdy=(y+\frac{xf\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$
$\Rightarrow \frac{dy}{dx}=\frac{y}{x}+\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)}$ which is homogenous
Putting $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$, we obtain
$v+x\frac{dv}{dx}=v+\frac{f\left(v\right)}{f^{\prime }\left(v\right)}\Rightarrow \frac{f^{\prime }\left(v\right)}{f\left(v\right)}dv=\frac{dx}{x}$
Integrating, we get log $f\left(v\right)=logx+logc$
$\Rightarrow logf\left(v\right)=logcx\Rightarrow f\left(\frac{y}{x}\right)=cx$