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Periodic Solution to FIrst Order ODE Proof

  1. Apr 14, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the differential equation x' = f(t,x) where f(t,x) is continuously differentiable in t and x. Suppose that

    f(t+T,x) = f(t,x) for all t

    Suppose there are constants p, q such that

    f(t,p) > 0, f(t,q) < 0 for all t.

    Prove that there is a periodic solution x(t) for this equation with p < x(0) < q.


    3. The attempt at a solution

    Not really sure what approach I'm supposed to take. I imagine I'm supposed to use the fact that f(t,x) is continously differentiable, but I'm not sure what that's supposed to give me. I wrote a few expressions using the fundamental theorem of Calculus, but that looked like a dead end. I'm not sure what I should copy here as previous work since I just tried a bunch of things that didn't seem to lead anywhere. If I could get a hint on what approach to use, that would be great.
     
  2. jcsd
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