What I'm trying to solve is
$ \frac{dx}{dy}=\frac{-x- y^3}{y - x^3}.$
I've tried with substituting with $ v = \frac{x}{y} $ and $ u = \frac{y}{x},$ but even with that I still can't separate the variables.
Using the second substitution I get $ ( u + x^3u^3 - x^2 + x ) dx + (u + u^4x^2) du = 0$
I tried working with this but found no way to simplify or advance from this in any way. I plugged it in Wolfram Alpha and of course it gave a nice looking solution to this (an ugly one for the first form of the equation though), so I know I'm missing something.
Any help would be greatly appreciated.