please answer the following questions using the method presented in the questions
AMATH 351 Autumn 2015
Due: Wednesday, December 09, 2015
1. ?The Empire Strikes Back?
? Go over the Exam for Part B (see .pdf ?le on CANVAS ? Files).
? Re-do all the questions you did poorly on or did not attempt. Show your work, the TA
will grade you on completeness.
? Train yourself to the point where you can solve all those questions correctly and in less
than 50 minutes. Don?t hate an exam you did poorly on and forget about it, on the
contrary, see it as an important learning experience. It is only by facing your fears that
your jedi training will be complete. 1 2. Find the Laplace transform of the function.
Use the de?nition of the Laplace transform.
s > |b| (a) f (t) = cosh bt,
(b) f (t) = eat cos bt, s>a 3. Laplace transform I
Use the Laplace transform to solve the initial value problem.
(a) y + 3y + 2y = 0, y(0) = 1, y (0) = 0 (b) y ? 4y + 4y = 0, y(0) = 1, y (0) = 1 (c) y + 2y + 5y = 0, y(0) = 2, y (0) = ?1 (d) y + 4y = 4t, y(0) = 1, y (0) = 5 4. The Heaviside function
Sketch a graph of the given functions on the interval t ? 0. uc (t) represents the Heaviside
(unit step) function.
(a) g(t) = u1 (t) + 2u3 (t) ? 6u4 (t)
(b) g(t) = sin(t ? 3)u3 (t)
Express f (t) in (c) f (t) = terms of uc (t) and sketch a graph of f (t).
7 ? t,
t?7 5. Laplace transform II
Use the Laplace transform to solve the initial value problem. For all problems, assume t ? 0.
(a) y + y = f (t), y(0) = 1, y (0) = 1, (b) y = ?(t ? t0 ), y(0) = 0, f (t) = 1, 0 ? t < 3?
0, t ? 3? t0 > 0 (c) y + 4y = ?(t ? ?) ? ?(t ? 2?), y(0) = 0, y (0) = 0 (d) y + 4y + 4y = ?(t ? ?/6) sin t, y(0) = 0, y (0) = 0 2
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