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
$\frac{xdv}{dx}+v=v-v^2$
$\Rightarrow \frac{xdv}{dx}=-v^2$
$\Rightarrow -\frac{dv}{v^2}=\frac{dx}{x}$
$\Rightarrow \frac{1}{v}=lnx+lnc$
$\Rightarrow \frac{x}{y}=lncx$
$\Rightarrow e^{\frac{x}{y}}=cx$.