1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted