[EricVanWyk]

Here are the simplifications I made:

1) Instantaneous perfect mixing.
2) flow in = flow out.
3) Track the amount of A in bucket A only. Don't bother with the other bucket or the other fluid.
4) Make the volume of the bucket 1 unit.

This simplifies things down to A' = -A; The amount of A that leaves the bucket is proportional to the amount of A left (times the amount of B entering).

This becomes A(B) = e^-B

or, pulling back from 1 unit to 5 gallons

A(B) = 5 gallons * e^-(B/5 gallons)