Solution of pendulum linked to Weierstrass $\wp$-function

Question

I've been working through a question about the equation of motion of a pendulum. I have to now solve the equation of the form: $$u'^2=u^3+au+b,$$
where $a=(\frac{g^2}{l^2}-\frac{c^2}{3})$ and $b=(\frac{g^2c}{3l^2}-\frac{2c^3}{27})$, using separation of variables.

So, I have separated the equation to get: $$\int\frac{1}{\sqrt{(u^3+au+b)}}\,\mathrm du=\int1\,\mathrm dt$$

Now, I could just integrate this directly and attempt a solution, but the next part of the question asks what the solution has to do with the Weierstrass $\wp$-function. Is there a trick we can use to find the solution?

Answer 1

Your answer is straight because your $u$ corresponds to a Weierstrass P-function. http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Differential_equation