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Math 0250 Exam 2Solutions:
November 6, 1998
F. Beatrous
 1.
 Find eigenpairs for the matrix
where a is a nonzero 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 c_{1} and c_{2} 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 x_{1} and x_{2} denote the amount of salt (in pounds) in the left and
right tank respectively. Then the salt concentrations in the two tanks are
x_{1}/100 and x_{2}/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
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Frank Beatrous
19981110