Math 215
Spring 2010
Assignment 5
You are highly recommended to own a textbook, Boyce-DiPrima or another, to see a comprehensive treatment of the subjects and more examples.
§3.8: 6, 8, 17(a,b,d),
§6.1: 10, 13, 21,
§6.2: 4, 10, 12, 23
• (§3.8: 6) A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external
force of 10 sin(t/2) N (newtons) and moves in a medium that imparts a viscous force
of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from
its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value
problem describing the motion of the mass.
• (§3.8: 8)
(a) Find the solution of the initial value problem in Problem 6 (the previous problem).
(b) Identity the transient and steady state parts of the solution.
(c) Plot the graph of the steady state solution.
(d) If the given external force is replaced by a force of 2 cos ωt of frequency ω, find
the value of ω for which the amplitude of the forced response is maximum.
• (§3.8: 17(a,b,d)) Consider a vibrating system described by the initial value problem
1
u00 + u0 + 2u = 2 cos ωt,
4
u(0) = 0,
u0 (0) = 2.
(a) Determine the steady state part of the solution of this problem.
(b) Find the amplitude A the steady state solution in terms of ω.
(d) Find the maximum value of A and the frequency ω for which it occurs.
• (§6.1: 10) Recall that cosh bt = (ebt + e−bt )/2 and sinh bt = (ebt − e−bt )/2. Find the
Laplace transform of f (t) = eat sinh bt, where a, b are real constants.
• (§6.1: 13) Recall that cos bt = (eibt + e−ibt )/2 and sin bt = (eibt − e−ibt )/2i. Assuming that the necessary elementary integration formulas extend to this case, find the
Laplace transform of f (t) = eat sin bt, where a, b are real constants.
Z ∞
• (§6.1: 21) Determine whether the integral
(t2 + 1)−1 dt converges or diverges.
0
• (§6.2: 4) Find the inverse Laplace transform of the function F (s) =
3s
.
s2 − s − 6
• (§6.2: 10) Find the inverse Laplace transform of the function F (s) =
2s − 3
.
s2 + 2s + 10
• (§6.2: 12) Use Laplace transform to solve the initial value problem
y 00 + 3y 0 + 2y = 0;
y(0) = 1,
y 0 (0) = 0.
• (§6.2: 23) Use Laplace transform to solve the initial value problem
y 00 + 2y 0 + y = 4e−t ;
y(0) = 2,
2
y 0 (0) = −1.