Help! Do not know how to solve ODE using homogeneity of differential equations : $xydx+(2x^2+3y^2)dy$

Question

What I did using exact equations

My attempt at homogeneity"

What am I doing wrong here math friends? Why can't I separate these variables?

Answer 1

Everything was fine until you've made a silly mistake. It should be
$$\large{v+x\frac{dv}{dx}=-\frac{v}{2+3v^2}}$$ $$\large{x\frac{dv}{dx}=-\frac{3v+3v^3}{2+3v^2}}$$ $$\large{\frac{2+3v^2}{3v+3v^3}}dv=-xdx$$

Answer 2

Integrating factor is $\mu=y^3$. Then solution of equation $$xydx+(2x^2+3y^2)dy=0$$ is $$y^6+x^2y^4=C$$