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Differential equation periodic and non periodic solutions.

  1. Sep 22, 2007 #1
    I have the next problem. Let be the linear differential equation [tex]\dot x=a(t)x[/tex] where [tex]a(t)[/tex] is a periodic function of fundamental period [tex]T>0[/tex],

    a) Give two examples: where the general solution is a periodic function again and where the general solution is a non-periodic function.

    b) Show that the general solution is periodic of fundamental period [tex]T>0[/tex] iff [tex]\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0[/tex].


    I have done yet the case when differential equation is [tex]\dot x=a(t)[/tex] and I have proof that solutions to this one are periodic iff [tex]\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0[/tex]. Can I use this fact?

    I also know that the general solution to the equation of my question is [tex]x(t)=ce^{\int_{}^{}a(s)ds}[/tex] and that the solutions to every IVP (except for that points over the equilibrium solution [tex]x(t)\equiv 0[/tex]) are [tex]x(t)=x_{0}e^{\int_{t_{0}}^{t}a(s)ds}[/tex] and that it implies that if this is periodic [tex]x_{0}e^{\int_{t_{0}}^{t}a(s)ds}-x_{0}e^{\int_{t_{0}}^{t+T}a(s)ds}=0[/tex]

    and [tex]x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t+T}a(s)ds}=x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t}a(s)ds-\int_{t}^{t+T}a(s)ds}=x_{0}e^{-\int_{t}^{t+T}a(s)ds}=0[/tex]

    but here I'm lost because it is that the integral couldn't exist but if [tex]a(t)[/tex] is periodic and that implies that integral exists. How proof that, and what functions are exaples of each type.
     
  2. jcsd
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