MA212: Assignment #6
Required Reading:.
• Section 10.4
To be turned in April 26th at the start of class.
Problems marked with ∗ require the use of maple.
1. Textbook: Section 10.4: 1, 4, 9, 24, 25, 28, 36, 37, 40
2. Textbook Chapter 10 review: 4, 12
3. In this problem you will solve a (relatively) simple problem in three ways, so that you can gauge the complexity of
each technique.
Consider a system with two tanks.
• Fluid flows into tank 1 at a rate r=5 L/min.
• Fluid flows out of tank 1 into tank 2 at a rate r=5 L/min.
• Fluid flows out of tank 2 at a rate r=5 L/min.
• Tank 1 has 100kg of Kool-Aid in it initially.
• Tank 2 has pure water in it initially.
• Tank 1 and 2 are filled with 500L of fluid initially.
• The fluid flowing into tank 1 is concentrated Kool-Aid which varies in concentration over time according to
f (t) = (1 + sin t)kg/L.
(a) Write down the matrix system of ODE describing this process. The system you write should be inhomogeneous,
x0 = Ax + f .
(b) Solve the homogeneous system, x0 = Ax, and write down its general solution.
(c) Solve the problem in part (a) using the method of undetermined coefficients, if possible.
(d) Solve the problem in part (a) using diagonalization, if possible.
(e) Solve the problem in part (a) using variation of parameters, if possible.
Note: I say “if possible” because we discussed in class that these methods can sometimes fail. If one or more of
these methods fails, explain why.
4.
∗
Four candidates are running for president, two from extremist political parties, and two from moderate parties.
Suppose voters switch between the candidates over time. The rate of switching is proportional to the number of
voters in that category. The switching rate constants (proportionality constants) are given in the diagram.
Voters for Extremist
1
^
1/2
1
Voters for@ Extremist 2
α
Voters for Moderate
1
k
1
1/2
+
Voters for Moderate 2
β
The rate constants α and β can be increased by a candidate giving an endorsement for the other.
1
(1)
(a) Write an homogeneous matrix differential equation, X 0 = AX, describing the number of voters supporting each
candidate. The vector X should be of the form


E1 (t)
 E2 (t) 

X(t) = 
M1 (t)
M2 (t)
where {E1 , E2 , M1 , M2 } are the number of voters of each type.
(b) Using only eigenvalues and eigenvectors, describe the vote distribution as t → ∞. Hint: all of your eigenvalues
are non-positive real numbers.
(c) Using your answer from the previous part, show that it’s not possible for either of the moderate candidates
to win the election. Then give a mathematical description of how they could adjust α and β to assure that
Extremist 1 wins.
2