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18.03 Problem Set 7
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.
III. Fourier series, Dirac delta function, and Laplace transform
L26 F 9 Apr Laplace transform: basic properties: EP 4.1.
L27 M 12 Apr Application to ODEs: SN 20; Notes H; EP 4.2.
R18 T 13 Apr Laplace transform.
L28 W 14 Apr Second order equations; completing the square; EP 4.3; SN 20.
R19 Th 15 Apr Laplace transform and differential equations.
L29 F 16 Apr The pole diagram: SN 22, 23.
L30 W 21 Apr Transfer function and complex gain.
R20 Th 22 Apr Review.
L31 F 23 Apr Hour Exam III
Part I.
26. (F 9 Apr) Let z be a complex number. Find, from the definition, L[ezt], and find
the region of convergence of the integral.
27. (M 12 Apr) Notes 3A-9 (Do (a) using the s-shift rule as suggested, but then do
it again by writing e−t sin(3t) as a linear combination of complex exponentials.); Notes
3B-1; EP 4.3: 1, 5.
28. (W 14 Apr) EP 4.3: 7, 8, 12, 13, 28, 36.
Part II.
26. (F 9 Apr) [Laplace transform] (a) Suppose that F (s) is the Laplace transform of
f(t), and let a > 0. Find a formula for the Laplace transform of g(t) = f(at) in terms
of F (s), by using the integral definition and making a change of variable. Verify your
formula by using formulas and rules to compute both L[f(t)] and L[f(at)] with f(t) = tn
.
(b) Use your 18.02 skills: Show that if h(t) = f(t) ∗ g(t) then H(s) = F (s)G(s). Do this
∞ ∞
by writing F (s) = f(x)e−sx dx and G(s) = g(y)e−sy dy; expressing the product
0 0
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