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Verhulst equation - integration problem

  1. Sep 23, 2011 #1
    Verhulst equation -- integration problem

    1. The problem statement, all variables and given/known data

    Integrate the unforced Verhulst equation over [itex]t[/itex] from [itex]t_{0}-\epsilon[/itex] to [itex]t_{0}+\epsilon[/itex], take the limit [itex]\epsilon \downarrow 0[/itex] and show this lead to the following requirement for the discontinuity in g:

    [tex]\stackrel{lim}{\epsilon \downarrow 0}[g(t,t_{0})]^{t_{0}+\epsilon}_{t_{0}-\epsilon}=F_{0}[/tex]

    2. Relevant equations

    Unforced Verhulst equation:
    [tex]\dot{g}-g+g^{2}=F_{0} \delta(t-t_{0})[/tex]

    Where:
    [tex]g(t)=\frac{1}{Ae^{-t}+1}[/tex]

    3. The attempt at a solution

    It looks to me that in order to solve this problem I need to show that:
    [tex]\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}g dt=\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}g^{2} dt[/tex]

    This will not be true of any function g in general, so we must show that it is true for the function in this problem. i.e. Show that:

    [tex]\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\frac{1}{Ae^{-t}+1} dt=\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\frac{1}{(Ae^{-t}+1)^{2}} dt[/tex]

    Is this along the right lines? This is where I get stuck as I'm not very well practiced in integration. I think I can do the one on the left, but am struggling with the one on the right.

    Thanks for any help.
     
  2. jcsd
  3. Sep 24, 2011 #2
    Re: Verhulst equation -- integration problem

    Any helpers out there?

    Incidentally (for those interested -- and in case you didn't already know) this equation is used in biology to model population dynamics. http://en.wikipedia.org/wiki/Logistic_function

    I'm doind an exercise on the application of Green's functions to nonlinear problem.
     
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