The solution of x dx + ydy = x²ydy – xy² dx is

$\begin{array}{l}\mathrm{xdx}+\mathrm{ydy}={\mathrm{x}}^2\mathrm{ydy}-{\mathrm{xy}}^2\mathrm{dx}\\ \mathrm{x}\left(1+{\mathrm{y}}^2\right)\mathrm{dx}+\mathrm{y}\left(1-{\mathrm{x}}^2\right)\mathrm{dy}=0\\ \frac{\mathrm{xdx}}{1-{\mathrm{x}}^2}+\frac{\mathrm{y}}{1+{\mathrm{y}}^2}\mathrm{dy}=0\\ \frac{-1}{2}\int \frac{-2\mathrm{xdx}}{1-{\mathrm{x}}^2}+\frac{1}{2}\int \frac{2\mathrm{y}}{1+{\mathrm{y}}^2}\mathrm{dy}=\log c\\ \Rightarrow \log \left(1+y^2\right)-\log \left|\left(1-x^2\right)\right|=\log c^2\end{array}$