First-order nonlinear ODE (approach is not clear to me)

I've been given the following ODE to be solved:

$\displaystyle (yx^2)y' + (2y^2x + 3x^3) = 0$

But I don't really know how to start. A direct integration is (obviously enough) not possible and it's not in the form to be solved by an integrating factor (or so I understand).

Then it could be solved by separation of variables (though I can't see what the separation itself would be), or by using the exact equation technique (though I'm not even sure how to check if it's in the appropriate form).

How do I get started?