I swear I have seen this type of ODE before, but I can't remember how to attack it. In general, I would like to know how to solve \left(f'(z)\right)^m = c\,G(z)^n where $m,\;n \in \mathbb{N}$ and $G(z)$ is just a polynomial in $f(z)$.
This sounds hard, so I would be happy with, \left(f'(z)\right)^2 = c\,G(z)^n, though this latter equation may be too difficult too. For my homework, though, I need to know, \left(f'(z)\right)^2 = \left( c\,f^3 + f^2 \right). It was also suggested in the homework question to utilize $g^2 = 3\,c-f.$ If I do this without thinking I get an equation \left(f'(z)\right)^2 = c\,f(z)^2, which seems much easier, though I am still a little rattled by the plus-minus.
In case it matters, this is a related to a method for solving the Korteweg-deVries equation, $u_t + u\,u_x + u_{xxx} = 0.$ I have seen some solutions (but did not understand them) where the polynomial was "factored" into 3 roots... something like that. I just don't want to know the answer, but how to get it. Please keep in mind that this is my first class in PDEs.
Thanks for any help!