Question

The solution of $ydx-xdy=xydx$ is:

• A. $y=c{x}^2$
• B. $y=c{x}^3$
• C. $x=cy{e}^x$
• D. $y=cx{e}^x$

Answer 1

Answer: (C)

$x=cy{e}^x$

$ydx-xdy=xydx$

$\Rightarrow y-x\frac{dy}{dx}=xy$

Divide with $-x^2$ on both sides

$\Rightarrow \frac{1}{x}\frac{dy}{dx}-\frac{y}{x^2}=\frac{-y}{x}$

$\Rightarrow \frac{d}{dx}\left(\frac{y}{x}\right)=-\left(\frac{y}{x}\right)$

$\Rightarrow \frac{d\left(y/x\right)}{\left(y/x\right)}=-dx$

$\Rightarrow log\left(\frac{y}{x}\right)-logc=-x$

$\Rightarrow log\frac{cy}{x}=-x$

$\Rightarrow \frac{cy}{x}=e^{-x}$

$\Rightarrow x=e^{+x}cy$