# Verhulst equation - integration problem

1. Sep 23, 2011

### billiards

Verhulst equation -- integration problem

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

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

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

2. Relevant equations

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

Where:
$$g(t)=\frac{1}{Ae^{-t}+1}$$

3. The attempt at a solution

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

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:

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

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. Sep 24, 2011

### billiards

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.