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3f-fall2010-exam_3_sample

# 3f-fall2010-exam_3_sample - Math 3F Exam#3 Practice Laney...

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Math 3F — Exam #3 Practice Laney College, Fall 2010 Fred Bourgoin Disclaimer: This might not cover all of the topics you are responsible for. And this practice exam seems too easy to me... Am I wrong? 1. Find the inverse Laplace transform of each function. (a) Y ( s ) = 1 - 2 s s 2 + 4 s + 5 (b) Z ( s ) = 2( s - 1) e 2 s s 2 - 2 s + 2 2. Use Maclaurin series (power series centered at 0) to solve the differential equation. Find the first four nonzero terms of each of the linearly independent solutions. (1 - x ) y ′′ + xy - y = 0 3. Find the Laplace transform of f ( t ) = t - u 1 ( t )( t - 1) for t 1. 4. Using the definition of the Laplace transform, find L{ e at cosh bt } . Hint #1: cosh x = 1 2 ( e x + e x ). Hint #2: The integral only converges when s - a > | b | . 5. Solve the initial-value problem. y ′′ + 3 y + 2 y = u 2 ( t ) , y (0) = 0 , y (0) = 1 6. Solve the initial-value problem. Your answer may be expressed in terms of an integral. y ′′ + 4 y + 4 y = g ( t ) , y (0) = 2 , y (0) = - 3 7. Use Laplace transforms to solve the initial-value problem. y ′′ + 2 y + 5 y = 0 , y (0) = 2 , y (0) = - 1 EC. Suppose F ( s ) = L{ f ( t ) } exists for s > a . Show that F ( n ) ( s ) = L{ ( - t ) n f ( t ) } , then use it to get L{ t 2 sin bt } .

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3f-fall2010-exam_3_sample - Math 3F Exam#3 Practice Laney...

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