# bernoulli equation

There's no Bernoulli ODE there, it's just a homogeneous ODE, since $xz\,\frac{dz}{dx}={{x}^{2}}+3{{z}^{2}}\implies \frac{dz}{dx}=\frac{x}{z}+\frac{3z}{x}.$
Put $z=yx,$ or having the initial form of your ODE put $y=z^2.$