1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 401

Exercise 9. a. Find the general solution to the homogeneo us system asso-

ciated with the nonhomogeneous system (18a) for a = 15 gal/min, (3 =

20 gal/min, / = 5 gal/min, 6 = 15 gal/min, V 1 = 200 gal , and V2 = 100 gal.

b. Find the general solution of the nonhomogeneous system (18a) for a, (3,

/, 6, Vi, and V2 as given in part a. and Ca= 1 lb/gal. (HINT: Assume there

is a particular solution of (18a ) in which each component is constant.)

c. Find the general solution of t he initial value problem (18) for a, (3, /,

6, Vi, and Vi, as given in part a., for Ca = 1 lb/min, and A 1 = 10 lbs and

Az = 0 lbs.

d. What is limt-+oo Q1(t)? What is limt...., 00 qz(t)? How does limt...., 00 q 1 (t)/V 1

·and limt_, 00 q2(t)/V2 compare with ca?

Exercise 10. Tank A is a 200 gallon tank and is initially filled with a brine

solution which contains 50 pounds of salt. Tank B is a 200 gallon tank and

is initially filled with pure water. Pure water flows into tank A from an

outside source at 5 gal/min. Solution from tank A is pumped into tank B at

7 gal/min. Solution from tank B is pumped back into tank A at 2 gal/ min

and out of the system at 5 gal/min. Find the amount of salt in each tank as

a function of time. What is the li miting value for the amount of salt in each

tank?

Exercise 11. Tank A is a 100 gallon tank and is initially filled with a brine

solution which contains 30 pounds of salt. Tank B is a 200 gall on tank and is

initially filled with a brine solut ion which contains 15 pounds of salt. A brine

solution containing 2 pounds of salt per gallon enters tank A from an outside

source at a rate of 5 gal/min. Solution from tank A is pumped into tank B at

a rate of 3 gal/min. Solution from tank B is pumped back into tank A at the

same rate. Solution from tank A is pumped out of the system at the rate of

5 gal/min. Find the amount of salt in each tank as a function of time. What

is the limiting value for the amount of salt in each tank?

Exercise 12. Three tanks are connected as shown in Figure 9.10. Each tank

contains 100 gallons of brine. The rates of flow are a = 10 gal/min , (3 =

5 gal/min, and I = 15 gal/min. Find the amount of salt in each tank as a

function of t ime, if initially tank 1 has 20 pounds, tank 2 has 5 pounds, and

tank 3 has 10 pounds. What is the limiting amount of salt in each tank?

I

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I

~ I I r=i

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a

CJ

Tan k 1 Tank 2 Tank 3

I --y I

Figure 9.10 Mixture Problem for Three Interconnected Tanks