Chapter 3.3, Problem 21E

Mixtures Solely on the basis of the physical description of the mixture problem on page 108 and in Figure 3.3.1, discuss the nature of the functions x₁(t) and x₂(t). What is the behavior of each function over a long period of time? Sketch possible graphs of x₁(t) and x₂(t). Check your conjectures by using a numerical solver to obtain numerical solution curves of (3) subject to the initial conditions x₁(0) = 25, x₁(0) = 0.

MIXTURES Consider the two tanks shown in Figure 3.3.1. Let us suppose for the sake of discussion that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t.

Figure 3.3.1 Connected mixing tanks

$\begin{array}{lll}\frac{d{x}_1}{dt}\hfill & =\hfill & -\frac{2}{25}{x}_1+\frac{1}{50}{x}_2\hfill \\ \frac{d{x}_2}{dt}\hfill & =\hfill & \frac{2}{25}{x}_1-\frac{2}{25}{x}_2.\hfill \end{array}$ (3)