MATH 215/255
Fall 2015
Assignment 5
§2.5.3, §2.6, §6.1
1 00
y + y 0 + y = e−2t sec(2t), − π4 < t < π4 .
4
Hint: To use the method of variation of parameters, rewrite the DE so that the
coefficient of y 00 is 1 before solving the boxed system on page 74 of [Lebl].
1. Find a particular solution of
2. Find the general solution of y 00 + 4y = 3 csc 2t,
0 < t < π/2.
3. Find the general solution of y 00 − 5y 0 + 6y = g(t). Here g(t) is an arbitrary continuous
function.
4. Verify that the given functions y1 and y2 satisfy the corresponding homogeneous
equation; then find a particular solution of the given nonhomogeneous equation.
ty 00 − (1 + t)y 0 + y = t2 e2t ,
t > 0;
y1 (t) = 1 + t,
y2 (t) = et .
Hint: To use the method of variation of parameters, first rewrite the equation as
1
0
2t
y 00 − 1+t
t y + t y = te ; then solve the same boxed system on page 74 of [Lebl].
5. A mass of 5 kg stretches a spring 9.8 cm. The mass is acted on by an external force
of 85 sin(2t) N (newtons) and moves in a medium that imparts a viscous force of 3
N when the speed of the mass is 5 cm/s. Suppose the mass is set in motion from its
equilibrium position with an initial velocity of 25 cm/s.
(a) Formulate the initial value problem describing the motion of the mass in MKS
units (meter/kilogram/second). Assume the gravity constant is 9.8 m/s2 .
(b) Find the solution of the initial value problem in (a).
(c) Identity the transient and steady periodic parts of the solution.
(d) If the given external force is replaced by a force of 85 cos ωt of frequency ω, find
the value of ω for which the amplitude of the forced response is maximum.
F0
.
Hint: By [Lebl, p.80], the amplitude is C = p
2
m (2pω) + (ω02 − ω 2 )2
6. Consider a vibrating system described by the initial value problem
1
x00 + x0 + 2x = 2 cos ωt,
4
x(0) = 0,
x0 (0) = 2.
(a) Determine the steady periodic part of the solution of this problem.
(b) Find the amplitude C of the steady periodic part of the solution in terms of ω.
(c) Find the maximum value of C and the frequency ω for which it occurs.
7. 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.
8. 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
9. Determine whether the integral
∞
(t2 + 1)−1 dt converges or diverges.
0
10. Determine the range of real s so that the integral converges:
Z ∞ 2t
Z ∞ 2t
e
e
e−st dt
(a)
e−st dt
(b)
2
1
+
t
1
+
t
0
0
2