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Math Help - No 2 Pi periodical solution of a ode

  1. #1
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    Wink 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<P(t)<(n+1)^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.
    Last edited by xinglongdada; July 26th 2011 at 04:23 PM.
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  2. #2
    Super Member girdav's Avatar
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    Re: No 2 Pi periodical solution of a ode

    Quote Originally Posted by xinglongdada View Post
    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<P(t)<(n+1)^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?
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  3. #3
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    Re: No 2 Pi periodical solution of a ode

    Yes...And I'm sorry that I neglected it. Now I correct the problem.
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  4. #4
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    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.
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