I have the following ODE
$$-\frac{vf^2}{2} = \frac{(f')^2}{2} + 2f^3 + Cf + D,$$
where $v, C, D$ are constants.
I've been given the hint to use elliptic functions (in particular, the Weierstrass P-function). I've looked at the wikipedia page for it, but really can't see how it can be of use.
Any help/hints?
HINT.-Clearly first you should know well the $\wp_L(z)$ for the lattice $L$. It is defined by
$$\wp(z)=\frac{1}{z^2}+\sum_{\omega\in L - \{0\}}\big [\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\big ]$$ and an important algebraic relation between the trascendental funtions $\wp$ and $\wp'$ is $$(\wp')^2=4(\wp)^3-g_2\wp-g_3\qquad (*)$$ where the called invariants of $\wp$, $g_2$ and $g_3$ are defined by $$g_2=60\sum_{\omega\in L - \{0\}}\frac{1}{\omega^4}\\g_3=140\sum_{\omega\in L - \{0\}}\frac{1}{\omega^6}\\$$ Now your equation has the form $$(f')^2=4(-f)^3-vf^2-2Cf-2D\qquad (**)$$
Try to associate the relations $(*)$ and $(**)$ in such a way that $f(x)=\wp'(t)$ and $x=\wp(t)$. You have to construct a lattice with $v,C,D$ and satisfying $C^3-27D^2\ne 0$. You can discard the term $vf^2$ in $(**)$ like as usual for solving the cubic equation (for a beginner all this it is not easy at all).