Question

Solve:
$xdy+ydx=x^{2}ydy$

Answer 1

$xdy+ydx=x^{2}ydy$

$\Rightarrow xdy+ydx={\left(xy\right)}^2\frac{dy}{y}$

$\Rightarrow \frac{d\left(xy\right)}{{\left(xy\right)}^2}=\frac{dy}{\left(y\right)}$ $(\because d(xy)=xdy+ydx)$

$\Rightarrow \frac{{\left(xy\right)}^{-2+1}}{(-2+1)}=(ln|y|+ln|c|)$ $c$=constant

$\Rightarrow \frac{-1}{xy}=ln\left(yc\right)|$

$\Rightarrow y=k{e}^{-1/xy}$ $k$=constant