# Find initial condition such that ODE solution is periodic

by TomAlso
Tags: condition, initial, periodic, solution
 P: 5 I have the following ODE system $\begin{cases} x' = v \\ v' = v - \frac{v^3}{3} - x \\ x(0) = x_0 \\ v(0) = 0 \end{cases}$ I am asked to find $x_0>0$ such that the solution $(x(t),v(t))$ is periodic. Also, I need to find the period $T$ of such solution. I don't know how to solve the system in the first place (or if it is even possible), so is there a way to figure out what $x_0>0$ will give a periodic solution without solving the system? Thanks!