# Thread: No 2 Pi periodical solution of a ode

1. ## No 2 Pi periodical solution of a ode

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 no non-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 the Sturm Comparision theorem. However, I could not derive any contradiction.

2. ## Re: No 2 Pi periodical solution of a ode

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 no $\displaystyle 2\pi$ periodical solution.
What about $\displaystyle x(t)=0$ for all $\displaystyle t$?