I have the following ODE system(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

\begin{cases}

x' = v \\

v' = v - \frac{v^3}{3} - x \\

x(0) = x_0 \\

v(0) = 0

\end{cases}

[/itex]

I am asked to find [itex]x_0>0[/itex] such that the solution [itex](x(t),v(t))[/itex] is periodic. Also, I need to find the period [itex]T[/itex] 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 [itex]x_0>0[/itex] will give a periodic solution without solving the system? Thanks!

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# Find initial condition such that ODE solution is periodic

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