This is actually related to a post I made earlier in the differential equations forums, but I've since realized that solving the equations themselves is not necessarily the best way to get where I want to go. Perhaps it's better suited to this forum, since it is an integration problem that I need to figure out.(adsbygoogle = window.adsbygoogle || []).push({});

So I need to calculate one of these integrals (one is enough, as it is then easy to get the others):

[itex]\int^{∞}_{0}[/itex]xydt or [itex]\int^{∞}_{0}[/itex]xdt or [itex]\int^{∞}_{0}[/itex]ydt

What I know is that x, y and t are related by the following equations:

dx/dt=-ax-cxy

dy/dt=-by-cxy

where a,b,c and the initial values x0 and y0 are all positive values. Because of this, both derivatives are negative while x>0 and y>0, and zero when x=0 and y=0 respectively. That means that it is clear that x(∞)=y(∞)=0.

The reason I'm posting this here is that, on hindsight, I do NOT need to solve the equations. I do not have any use for the functions x(t) or y(t). I only need one of the integrals mentioned above.

I have managed to find some non-elementary solutions, but I'm still not quite convinced that the integral can't be expressed with elementary functions. I know it CAN be when a=b, but not sure when a≠b.

I'm just thinking that I'm overlooking some relatively simple way to do this. Has anyone come across an integration problem like this before?

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# Integrating directly from a pair of diff.eqs. without solving them

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