Solution of the differential equation

$y\left(x y+2 x^2 y^2\right) d x+x\left(x y-x^2 y^2\right) d y=0$

is given by

$2 \log |x|-\log |y|-\frac{1}{x y}=C$

We have,

$\left(x y^2+2 x^2 y^3\right) d x+\left(x^2 y-x^3 y^2\right) d y=0$

$\Rightarrow x y(y d x+x d y)+x^2 y^2(2 y d x-x d y)=0$

$\Rightarrow \frac{d(x y)}{x^2 y^2}+\left(\frac{2}{x} d x-\frac{1}{y} d y\right)=0$         [Dividing by $x^3 y^3$]

On integrating, we get

$-\frac{1}{x y}+2 \log |x|-\log |y|=C$