1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 399

which the solution enters the container from the other container minus the

concentration of the substance in the container times the rate at which the

solution leaves the container, we obtain the following system of differential

equations:

dq1 qz(t) q1(t)

• = c2(t)r - c1 (t)r = r-- - r--
  dt Vz Vi

dqz = c1(t)r - c2(t)r = r qi(t) - r qz(t).

dt Vi Vz

Rewriting this system in matrix-vector notation, we see q 1 and q 2 satisfy

q'~ C) (~ ~:)CJ

V1 Vz

(16a)

The initial conditions are q 1 (0) = A 1 and q 2 (0) = A 2 or

(16b)

(

q1 (0)) (A1)

q(O) = =.

qz(O) Az

For our next example, suppose two containers are connected by tubes of

negligible length as shown in Figure 9.9. Suppose at time t = 0 an amount

A 1 of a substance is present in a solution that fills a container of constant
volume Vi and an amount Az of the same substance is present in a solution
that fills a container of constant volume Vz.

~

a----+ ----+

V1

D

V2

+---- ----+8

y

Figure 9.9 Mixture Problem for Two Interconnected Tanks

Also assume at time t = 0 that (i) a solution containing a concentration Ca

of the substance is allowed to enter the first container from an outside source
at the rate a; (ii) the solution in the first container is kept at a uniform
concentration c 1 ( t) and is pumped into the second container at a constant
rate /3; and (iii) the solution in the second container, which is kept at a
uniform concentration c 2 (t), is pumped back into the first container at the

constant rater and is pumped out of the system at the constant rate 8. We