# 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
$\frac{d^2x}{dt^2}+P(t)x=0,$
where $P(t)$ is continuous function of $t$ with period $2\pi$, and satisfies $n^2. Here $n$ is a non-negative integer. Prove then this ode has no non-trivial $2\pi$ periodical solution.

There was a hint about this problem: Argue by contradiction by assuming there is a non-trivial $2\pi$ 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

Originally Posted by xinglongdada
Consider the ode $\frac{d^2x}{dt^2}+P(t)x=0,$
where $P(t)$ is continuous function of $t$ with period $2\pi$, and satisfies $n^2. Here $n$ is a non-negative integer. Prove then this ode has no $2\pi$ periodical solution.
What about $x(t)=0$ for all $t$?

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

Yes...And I'm sorry that I neglected it. Now I correct the problem.

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

Perhaps I've got the idea. To prove that during the period of the solution, there was exactly an odd number of zeros.