(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Verhulst equation - integration problem

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