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

$\displaystyle \frac{d^2x}{dt^2}+P(t)x=0,$

where $\displaystyle P(t)$ is continuous function of $\displaystyle t$ with period $\displaystyle 2\pi$, and satisfies $\displaystyle n^2<P(t)<(n+1)^2$. Here $\displaystyle n$ is a non-negative integer. Prove then this ode has nonon-trivial$\displaystyle 2\pi$ periodical solution.

There was a hint about this problem: Argue by contradiction by assuming there is a non-trivial $\displaystyle 2\pi$ periodical solution and apply theSturm Comparision theorem. However, I could not derive any contradiction.