Subject: Integration Constant
Date: Thu 12/24/98 6:05 PM
From: Jim

Alexander:

Perhaps you can help me understand one step in the solution of the following problem involving the system of differential equations. A solution flows from one container into a second container at a rate proportional to the volume of solution in the first container. It flows out of the second container at a constant rate. Let x(t) = volume in container 1; y(t) = volume in container 2.

EQ 1:  dx/dt = -ax
EQ 2:  dy/dt = ax - b, where a, b are constants

Integrating the first equation, x(t) = x(0)e-at Substituting this into EQ 2 above: dy/dt = ax(0)e-at - b. Upon Integration, yields the solution: y(t) = y(0) + x(0)(1 - e-at) - bt.

This last step is unclear. It looks like the constant of integration, let's call it c, is assumed to equal: c = y(0) + x(0). I can see that the solution makes sense (except for my note below), but how do you know to split c into x(0) and y(0). Also, how do you handle the problem where t=infinity, so the term bt goes to infinity?

Regards,
Jim

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