Due by 12:45 P.M., Friday, March 14, 2008, in the boxes at 2106, next to the Under
graduate Mathematics Oﬃce.
I encourage collaboration on homework in this course. However, if you do your homework
in a group, be sure it works to your advantage rather than against you. Good grades for
homework you have not thought through will translate to poor grades on exams.
You
must turn in your own writeups of all problems, and, if you do collaborate,
you must write on the front of your solution sheet the names of the students
you worked with.
Because the solutions will be available immediately after the problem sets are due,
no
extensions will be possible
.
L13
W 5 Mar
Driven systems: Superposition, sinusoidal harmonic response:
Notes O.1; EP 2.6, pp. 157–159 only—see SN 7 if you want
to learn about beats.
R9
Th 6 Mar
Complex numbers and phase lag.
L14
F 7 Mar
Operators, exponential response formula:
SN 10, Notes O.1, 2, 4, EP 2.6 (pp. 165–167).
L15
M 10 Mar
Undetermined coeﬃcients; RLC circuits:
SN 11, EP 2.5 (pp. 144–153); EC 2.7, SN 8.
R10
T 11 Mar
Operators, ERF, undetermined coeﬃcients.
L16
W 12 Mar
Frequency response: SN 14.
R11
Th 13 Mar
Ditto.
L17
F 14 Mar
Linear time invariant operators.
Part I.
13. (W 5 Mar) (a)
[From R8.] Two facts about solutions to
m
¨
x
+
b
˙
x
+
kx
= 0: (i) The
real part of a solution is again a solution. (ii) Any constant multiple of a solution is again
a solution. So if
e
rt
is a solution (with
r
=
a
+
ci
a complex root, say), then so is
z
0
e
rt
,
for any complex number
z
0
. If we write
z
0
=
Ae

φi
(so
A
=

z
0

and
φ
=

Arg (
z
0
)),
what is the real part of
z
0
e
rt
? Explain why this gives the general solution. What would
you take for
z
0
to get Im
e
rt
?
(b)
For each of the following functions
f
(
t
), ﬁnd a complex number
z
0
and a positive
real number
ω
such that
f
(
t
) = Re (
z
0
e
iωt
). (i) cos(2
t
). (ii) 2 sin(
πt
). (iii) cos(
t
+
π
4
). (iv)
cos(
t
) + 3 sin(
t
).
(c)