Dear all, I've got a question as follows. I thought about it for more than two days, but could not derive any contradiction. Would you help me?

Consider the ode

where is continuous function of with period , and satisfies . Here is a non-negative integer. Prove then this ode has nonon-trivialperiodical solution.

There was a hint about this problem: Argue by contradiction by assuming there is a non-trivial periodical solution and apply theSturm Comparision theorem. However, I could not derive any contradiction.