Question

If $\frac{xdy}{dx}+\frac{y^{2}}{x}=y$, then

• A. $e^{x/y}=cx$
• B. $e^{y/x}=cx$
• C. $e^{x/y}=cy$
• D. $e^{y/x}=cy$

Answer 1

Answer: (A)

$\frac{dy}{dx}=\frac{xy-y^2}{x^2}$
Substitute $y=vx $
$\Rightarrow \frac{dy}{dx}=\frac{xdv}{dx}+v$
Now, given equation becomes
$\frac{xdv}{dx}+v=v-v^{2}$
$\Rightarrow \frac{xdv}{dx}=-v^{2}$
$\Rightarrow -\frac{dv}{v^{2}}=\frac{dx}{x}$
$\Rightarrow \frac{1}{v}=ln\,x+ln\,c$
$\Rightarrow \frac{x}{y}=ln\,cx$
$\Rightarrow e^{\frac{x}{y}}=cx$.