Math 0250 Exam 2Solutions:

November 6, 1998

F. Beatrous

1.
Find eigenpairs for the matrix

where a is a non-zero number.

Solution:
The characteristic polynomial is

so the eigenvalues are 0 and 2a.

For the eigenvalue ,

which gives the eigenvector

and the eigenpair

For the eigenvalue ,

which gives the eigenvector

and the eigenpair

2.

(a)
The matrix

has eigenvalues 3 and -1 with corresponding eigenvectors

Find the general solution to the system X'=AX.

Solution:

(b)
Solve the initial value problem

Solution:
Substituting the initial condition into the general solution from 2a gives

Solving for c1 and c2 gives

so the required solution is

3.
(a)
With A as in problem 2, find all solutions to the system

Solution: Look for a particular solution of the form X=Ce-2t. Substituting into the above system gives

and canceling the exponential factors gives

and rearranging gives

Solving gives

so the required particular solution is

The general solution is obtained by adding the complementary solution from problem 2a:

(b)
Find the solution to the system of part 3a satisfying the initial condition

Solution:
Substituting the initial condition into the general solution form 3a gives

and solving gives

so

4.
Two 100 gallon tanks are connected by pipes as indicated. Both tanks are initially filled with pure water. Starting at time t=0 minutes, a brine solution containing 10 pounds of salt per gallon of solution is pumped into the left tank at a rate of 10 gallons per minute. Fluid is pumped from the left tank to the right tank at a rate of 15 gallons per minute, and from the right to the left at a rate of 5 gallons per minute. Fluid is allowed to drain from the right tank at a rate of 10 gallons per minute. Set up a system of differential equations and initial conditions for the amount of salt in each tank. What is the coefficient matrix for the system? What is the forcing function? Do not solve the system.

Solution: Let x1 and x2 denote the amount of salt (in pounds) in the left and right tank respectively. Then the salt concentrations in the two tanks are x1/100 and x2/100. The rate of change of the amount of salt in the left tank is

Similarly, the rate of change in the right tank is

In matrix notation,

Since the initial amount of salt in both tanks is zero, the initial condition is

The coefficient matrix is

and the forcing function is