1. Consider a cascade of two tanks where a solution with time-varying concentration m(t)
flows into tank 1 at a rate of 0.5 gal/min, the well-mixed solution in tank 1 flows into
tank 2 at the same rate, and further the well mixed solution in tank 2 drains out at the
same rate. Suppose the concentration of salt m(t) is 12 for 0 < t < 3 and 0 for t > 3,
initially tank 1 contains 20 gallons of fresh water, and tank 2 contains 10 gallons of a
solution with concentration 1 lb/gal.
• Set up an initial value problem to model the above system of salt tanks using the
amounts of salt x1 (t) and x2 (t) in tanks 1 and 2 respectively.
• Solve the system, and find the amounts of salt in each tank as a function of t.
• Sketch the function representing the amount of salt in tank 2 and estimate after
what time (if ever) the amount of salt in tank 2 is guaranteed to be less than 1
lb.
2. A linear spring-mass-dashpot system has mass 2 kg, spring constant 10 Newtons per
meter, and damping constant of 4 Newtons per meter per second. Let x(t) denote the
position of the mass at time t with x = 0 denoting equilibrium. At time t = 0 the
mass is at rest at its equilibrium position. At time t = 1 an impulse blow with total
impulse 3 sets the mass in motion.
• Model the situation mathematically using a second order DE.
• Solve the DE using the method of Laplace transforms.
• What is the first time after t > 1 at which the mass returns to the equilibrium
position? What is its velocity at that moment? Suppose that at that instant
we can deliver an impulsive force to the mass. What total impulse should be
imparted to bring the mass exactly to rest?
3. Consider a constant population of P individuals. A certain disease infects this population. Infected individuals may (or may not) recover, but a certain fraction of those
who do recover remain susceptible and may again become re-infected. Let s(t) be
the number of susceptible individuals in the population and u(t) the number of those
currently infected. The SIR model for this situation is given by the nonlinear system
of DE’s
s = −bus + g(P − s − u)
u = bus − ru
where P is the total population, 0 < r < 1 is the fraction of the infected individuals
who recover, 0 < g < 1 is the fraction of infected individuals that may again be
infected, and b is a certain constant of proportionality (which governs how easily the
disease is spread). (Note: the strict < should be less than or equal - my printer has
trouble with the less than or equal symbol - RGL)
Consider the case in which r = 0.9, g = 0.2, b = 0.01, and P = 1000.
• Find the fixed points (critical points) for this system.
• Linearize about each fixed point and determine its stability characteristics.
• Write a plain English sentence describing the physical meaning of each fixed point.
4. Suppose we are given the function f (t) such that f(0) = 0, f(2) = 0, f(4) = −1,and
t
,
2
0<t<2
−1 2 < t < 4
f (t) =
4.a Let E(t) be the Fourier series of the even periodic extension of f (t) with period 8.
Plot E(t) for −8 < t < 8. (as before, these less than’s (same in (c) should be less than
or equal)
4.b What are the values for E(4), E(8), E(−12)?
4.c Let O(t) be the Fourier series of the odd periodic extension of f(t) of period 8. Plot
O(t) for −8 < t < 8.
4.d What are the values for O(4), O(8), O(−12)?
5. Let the function f be defined by
f (t) =
−t, −2 < t < 0
t,
0<t<2
f (t + 4) = f (t).
• Compute the Fourier series of f.
• Determine how many (non-zero) terms are needed in the Fourier series so that
the error is less than 0.01 for all t.