18.03 Problem Set 2
Due by 12:45 P.M., Friday, February 22, 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
.
I. Firstorder diﬀerential equations
L4
W 13 Feb
Solution of linear equations; integrating factors:
EP 1.5, SN
§
3.
R4
Th 14 Feb
Ditto
L5
F 15 Feb
Complex numbers, roots of unity: SN 5–6; Notes C.1–3.
L6
T 19 Feb
Complex exponentials; sinusoidal functions: Notes C.4; SN 4; Notes IR.6.
L7
W 20 Feb
Linear system response to exponential and sinusoidal input;
gain, phase lag: SN 4, Notes IR.6.
R5
Th 21 Feb
Complex numbers and exponentials.
L8
F 22 Feb
Autonomous equations; the phase line, stability: EP 1.7, 7.1.
Part I.
4. (W 15 Feb)
EP 1.5: 1, 2, 5. [Remember to check to see if the equation is separable
ﬁrst.]
5. (F 17 Feb)
Notes 2E1, 2, 7.
6. (T 21 Feb)
Notes 2E9, 10. Write each of the following functions
f
(
t
) in the form
A
cos(
ωt

φ
). In each case, begin by drawing a right triangle with sides
a
and
b
. (a)
cos(2
t
) + sin(2
t
). (b) cos(
πt
)

√
3 sin(
πt
). (c) cos(
t

π/
8) + sin(
t

π/
8).
7. (W 22 Feb)