Recently I am facing a problem which leads to the following type of ODE $$x'' + x + f(x') = 0.$$
It is obvious that when $f$ is a constant valued function, the solutions are all periodic and have a common period $2\pi$.
I want to know if there is a nonconstant smooth function $f$, such that (1) the solutions are all periodic, and (2) they share a common period $T$.
I have obtained many examples which satisfy the condition (1), for example $f(x)=(\sin x)^n$ for even integer $n>0$. But none of them satisfy (2).