Question

The general solution of the differential equation $xdy-ydx=y^{2}dx$ is

• A. $y=\frac{x}{C-x}$
• B. $x=\frac{2y}{C+x}$
• C. $(C+x)\left(2x\right)$
• D. $y=\frac{2x}{C+x}$
• E. $x=\frac{y}{C-x}$

Answer 1

The correct option is A $y=\frac{x}{C-x}$

We have $xdy-ydx=y^{2}dx$
$\Rightarrow x\frac{dy}{dx}-y=y^2$
$\Rightarrow \frac{x\frac{dy}{dx}-y}{y^2}=1$
$\Rightarrow -\left(\frac{y-x\frac{dy}{dx}}{y^2}\right)=1$
$\Rightarrow -\frac{d}{dx}\frac{x}{y}=1$
$\Rightarrow \frac{x}{y}=-x+C$
$\Rightarrow y=\frac{x}{C-x}$