1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 491

EXERCISE 10.9

l. Numerically solve system (2) on the interval [O, 10] for (a) E = .5 and

(b) E = 2 for the initial conditions Y2 (0) = 0 and (i) y 1 (0) = .5 , (ii) y 1 (0) = 2,

and (iii) Y1 (0) = 3. In each case, display a phase-plane graph of y 2 versus

Yl· Notice in cases (a) and (b) there is a periodic solution to van der Pol's

equation. This periodic solution is called a limit cycle, since solutions which

begin inside the periodic solution "spiral outward" toward the periodic solu-

tion and solutions which begin outside the periodic solution "spiral inward"

toward the periodic solution.

10.10 Mixture Problems

We introduced and studied mixture problems in chapters 3 and 9. Recall

the following underlying assumptions for solving mixture problems:

The rate of change of the amount of substance in any container at time

t is equal to the sum over all inputs to the container of the concentration

of each input times the rate of input minus the concentration of the

substance in the particular container times the sum of the rates of flow

of outputs from the container.

EXAMPLE 13 A Three Pond Mixture Problem

Three small ponds containing 200 gallons of pure water are formed by a
spring rain. Water containing .73 lbs/gallon of salt enters pond A at the
rate of 100 gal/hr. Water evaporates from pond A at the rate of 30 gal/hr
and flows into pond B at 70 gal/hr. Water evaporates from pond B at
20 gal/hr and flows into pond C at 50 gal/hr. Water evaporates from pond
C at 25 gal/hr and flows out of the system at 25 gal/hr. Find the amount of
salt in each pond as a function of time. What is the limiting amount of salt
in each pond?

SOLUTION

Let y 1 (t) be the amount of salt in pond A at time t. The concentration of

salt in the inflow to pond A is Ci = .73 lbs/gal and the inflow rate is ri =

100 gal/hr. Since the evaporation rate plus the outflow rate is also 100 gal/hr,
the volume of pond A remains constant. Hence, the concentration of salt in
the outflow from pond A to pond Bat any time is c 0 = y 1 (t)/200 (lbs/gal).
Applying our underlying principle to pond A, we find