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

[zeroseven]

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.

So I need to calculate one of these integrals (one is enough, as it is then easy to get the others):
$\int^{∞}_{0}$xydt or $\int^{∞}_{0}$xdt or $\int^{∞}_{0}$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?

 

[Mark44]

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.

So I need to calculate one of these integrals (one is enough, as it is then easy to get the others):
$\int^{∞}_{0}$xydt or $\int^{∞}_{0}$xdt or $\int^{∞}_{0}$ydt

What I know is that x, y and t are related by the following equations:
dx/dt=-ax-cxy
dy/dt=-by-cxy

You can't evaluate any of the three integrals directly. Both x and y are functions of t, but you aren't given that information.

All three of your integrals are with respect to t (indicated by dt in the integrand). Unless you can rewrite x, y, or both explicitly in terms of t, you can't evaluate the integral.

IOW, we can do this integration: ∫x dx = (1/2)x² + C.
We can't do this integration: ∫x dt, without knowing how x and t are related.

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?

 

[Mute]

It's not immediately clear to me if you can solve for the integrals without actually solving the differential equation. The relationships between the three integrals implied by the differential equation may not be enough.

One thing that you might be able to exploit is the symmetry $(x,y,a,b) \rightarrow (y,x,b,a)$, which your system of differential equations exhibits.

If we let

$$X(x_0,y_0,a,b,c) = \int_0^\infty dt~x(t),$$
$$Y(x_0,y_0,a,b,c) = \int_0^\infty dt~y(t),$$
$$Z(x_0,y_0,a,b,c) = \int_0^\infty dt~x(t)y(t),$$

then it follows from that symmetry that

$$X(x_0,y_0,a,b,c) = Y(y_0,x_0,b,a,c)$$
and
$$Z(x_0,y_0,a,b,c) = Z(y_0,x_0,b,a,c).$$

(It of course also follows that $Y(x_0,y_0,a,b,c) = X(y_0,x_0,b,a,c)$).

I'm not sure if you can say anything more than that without solving the DE's. You may be able to squeeze out some extra results by playing around with those symmetries and the system of equations for X, Y and Z, but you may not be able to pin down the values explicitly.