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Show all of your work. Full credit may not be given for an answer alone. This is a non-calculator exam. You may use the backs of the pages for scratch work. Do not unstaple or remove pages.

By taking this exam you are agreeing to abide by Rutgers University's Academic Integrity Policy.

 Problem Possible Points Number Points Earned 1 15 2 15 3 15 4 20 5 15 6 20 Total 100

1. (15 points)
1. (10 points) Compute , using only the definition of Laplace transform.
2. (5 points) Compute , using either the definition or by differentiating part (a).

2. (15 points) Compute .

Hint 1   .

3. (15 points) Use the Laplace transform to solve the ordinary differential equation

4. (20 points) Let and suppose is a solution to the ordinary differential equation

1. (15 points) Show for some constant .
2. (5 points) Suppose . What is in terms of ?

5. (15 points) Let on . Calculate the Fourier coefficients of and write down the Fourier series.

6. (20 points) Let and define on . Calculate the Fourier coefficients of and write down the Fourier series.

Matthew Leingang 2002-10-10