Consider the non-autonomous equation $$\ddot x + x = \epsilon f(x,\dot x) + \epsilon \sin(3t)$$ Assume that this equation has (a) periodic solution(s). What are the possible periods?
I need to assume that this equation has (a) period solution(s), e.g., solutions $x = \phi(t)$ for which $\phi(t + T) = \phi(t)$ for some $T$ and I need to find for which $T$ this is possible. As a start I entered $\ddot x + x = \epsilon\sin(3t)$ ($f(x,\dot x) = 0$) in Wolfram for different $\epsilon$ and it seems as if $\epsilon$ doesn't have an impact on the period of the solution for this equation. I therefore think that the possible periods depend on the function $f(x,\dot x)$, but I'm not really sure which options for $f(x,\dot x)$ I should consider. Furthermore, I think that I should do something with $\sin(3t)$ since I think this term will definitely have an impact on the possible periods.
My Question: I really have no clue how I should solve this problem, and I'm not necessarily looking for the direct solution. I would like to know where to start and which questions I should ask so that I can find the solution by myself.
Thanks in advance!