I've met the following problem in finishing my argument. The expression I ended with is $\frac{\mathrm d}{\mathrm d\xi}U(\xi)=\pm\sqrt{C_1-\frac{U^4(\xi)}{2}},$ with $U:\mathbb R\to\mathbb R$ and $C_1$ is non zero because otherwise I would have no square root unless $U\equiv 0$, which is not admissible in my situation. Therefore I would say the solution $U$ is periodic, and also I would like to find some bounds on the period.
The first problem I ask is the following: how would you proceed in showing that $U$ is periodic?
The second question is to find some bounds on the period. I mean: separating the variables and choosing the $+$ sign one gets $\frac{\mathrm d U}{\sqrt{C_1-\frac{U^4}{2}}}=\mathrm d \xi,$ but then I derived no useful informations about the period. Could you help me?
Thank you in advance.