1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 493

The amount of salt in pond A begins to increase first. The amount of salt

in pond B begins to increase next and finally the amount of salt in pond

C begins to increase. Equating the right-hand-sides of the three differential

equations in system (1) to zero, we find the critical point of system (1) to be

(208.57, 292, 584). This critical point is asymptotically stable, so as t _, oo,

Y1 -> 208.57, Y2 -> 292 and y3 -> 584.

Pollution in the Great Lakes In chapter 9 we derived the following

equations for the concentration of pollution in the Great Lakes. (See equa-

tion 19 and the accompanying table of constants in chapter 9.)

(2)

I 15C1 - 15y1

Yi = 2900

I 38C2 - 38y2

y^2 = 1180

15C3 + 15y1 + 38y2 - 68y3

y~ = -----------

850

17C4 + 68y3 - 85y4

y~ = --------

116

14Cs + 85y4 - 99ys

y~ = 393

Here Yi ( t) is the concentration of pollution in lake i at time t and Ci is the con-
centration of pollutant in the inflow to lake i. Subscript 1 corresponds to Lake
Superior, subscript 2 corresponds to Lake Michigan, subscript 3 corresponds
to Lake Huron, subscript 4 corresponds to Lake Erie, and subscript 5 corre-
sponds to Lake Ontario. In system (2) the unit of measure of the independent
variable, x (time), is years.

EXERCISE 10.10

1. Solve system (2) on the interval [O, 200] assuming all the Great Lakes have
   the same initial pollution concentration of Yi(O) = .5% and

a. the inflow concentration of pollutant for all lakes is reduced to Ci = .2%.

b. the inflow concentration of pollutant for all lakes is reduced to zero-
Ci = 0.

(Enter Yi(O) as .5 and enter Ci as .2 or 0, so your results will be expressed in
percent. Otherwise, you may experience some numerical difficulties.)