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(solution) please answer the following questions using the method presented

please answer the following questions using the method presented in the questions

AMATH 351 Autumn 2015

Homework 7

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).

t,

2,

7 ? t,

0, 0?t<2

2?t<5

5?t<7

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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