- #1
billiards
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Verhulst equation -- integration problem
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}
Unforced Verhulst equation:
\dot{g}-g+g^{2}=F_{0} \delta(t-t_{0})
Where:
g(t)=\frac{1}{Ae^{-t}+1}
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.
Homework Statement
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}
Homework Equations
Unforced Verhulst equation:
\dot{g}-g+g^{2}=F_{0} \delta(t-t_{0})
Where:
g(t)=\frac{1}{Ae^{-t}+1}
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.
