1550078481-Ordinary_Differential_Equations__Roberts_

400 Ordinary Differential Equations

assume both containers are always full. This means the sum of the input rates

to a container must equal the sum of the output rates. Hence, for the system

under consideration the following relationships between the rates of flow must

hold:

(17) a+'Y=f3

(for container 1)

f3='Y+o

(for container 2)

Let qi (t) be the amount of substance in the first container and q2(t) be the

amount of substance in the second container at time t. The concentration of

the substance in the first container is ci ( t) = qi ( t) /Vi and the concentration

of the substance in the second container is c 2 (t) = q 2 (t)/Vi. Equating the

rate of change of the amount of substance in each container to the sum over

all inputs to the container of the concentration of the input times the rate

of input minus the concentration of the substance in the container times the

sum of the rates of output from the container, we obtain the following system

of differential equations for the mixture system under consideration:

Or, writing this system in matrix-vector notation

The initial conditions for this system are

(18b)

Exercise 8. a. Use EIGEN or your computer software to find the general

solution to system (16a) for Vi = 100 gal, V 2 = 50 gal, and r = 10 gal/min.

b. Solve the initial value problem (16) for Vi, Vi, and r as given in part a.,

Ai = 50 lbs, and A2 = 0 lbs. That is , determine values for the arbitrary con-
stants in the general solution for part a. to satisfy t he initial conditions (16b)

when Ai = 50 lbs and A2 = 0 lbs.

c. What is limt-><xi qi (t)? What is limt_, 00 q2(t)? Are these values the

values you expect from strictly physical considerations?