The equation (3x²y + 2) dx + (x³ + 2y) dy = 0 is exact…

$M\,dx + N\,dy = 0$ is exact when $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$, and then $F(x, y) = C$ with $F_x = M$ and $F_y = N$. If $\dfrac{M_y - N_x}{N}$ depends on $x$ only, $\mu = e^{\int \frac{M_y - N_x}{N}dx}$. If $\dfrac{N_x - M_y}{M}$ depends on $y$ only, $\mu = e^{\int \frac{N_x - M_y}{M}dy}$. For homogeneous $M$, $N$ use $\mu = \dfrac{1}{Mx + Ny}$.