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421s08x1h - Math 421:01&02 Prof Bumby Exam 1 Name All class...

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Math 421:01&02 Prof. Bumby Exam 1 February 20, 2008 Name All class exams will be graded on the basis of 100 points. On this exam, those points are distributed over 7 problems (on 7 pages). No books or papers other than the official formula sheet may be used, but the formulas on that sheet may be used to find any quantity needed for your work. Calculators or other electronic devices are not allowed. You may leave when you are finished. 1. (15 pts.) Find the following Laplace transforms: a . ˇ ˚ e 5t cos 2t D . b . ˇ ˚ sin 2 t D . c . ˇ ˚ .t 3/ 2 D .
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Math 421:01&02, February 20, 2008, p. 2 2. (15 pts.) Find the following inverse Laplace transforms: a . ˇ 1 ˚ 6s 3 s 2 C 9 D . b . ˇ 1 ˚ 3s C 29 .s 2/.s C 3/ D . c . ˇ 1 ˚ 144 s 2 .s 4/ D .
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Math 421:01&02, February 20, 2008, p. 3 3. (15 pts.) Solve the initial value problem y 0 C y D 51e 3t cos t; y.0/ D 20:
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Math 421:01&02, February 20, 2008, p. 4 4. (20 pts.) Let f .t/ D 8 < : sin t if 0 t < =2 cos t if t =2 G.s/ D 3 s 2 C 4e 3s s a . Write f .t/ using an appropriate Heaviside function. b . Use the result of part a to find the Laplace transform F.s/ . c . Express the inverse Laplace transform g.t/ using Heaviside functions.
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