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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 |

- (15 points)
- (10 points)
Compute
, using only the definition of Laplace
transform.
*Proof*. [Solution]

- (5 points)
Compute
, using either the definition or by differentiating
part (a).
*Proof*. [Solution]One can do integration by parts to find , but the easiest way is with differentiation.

- (10 points)
Compute
, using only the definition of Laplace
transform.
- (15 points)
Compute
.
*Hint 1*.*Proof*. [Solution]Using the hint, we have

So - (15 points)
Use the Laplace transform to solve the ordinary differential equation
*Proof*. [Solution]The Laplace transform of the equation is

We factor and thus we have a partial fraction decomposition

This solves to give , . Thus

- (20 points) Let
and suppose is a solution to the
ordinary differential equation
- (15 points) Show
for some constant
.
*Proof*. [Solution]Remember that transforms to , so the transformed equation is

The integrating factor for this ODE in is

Thus

- (5 points) Suppose . What is in terms of ?

*Proof*. [Solution]Notice that - (15 points) Show
for some constant
.
- (15 points)
Let
on
. Calculate the Fourier coefficients of
and write down the Fourier series.
*Proof*. [Solution]This function is even, so all of its sine Fourier coefficients are zero. So we need only integrate between 0 and , and count it twice as much. Moreover, on , . So

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

Assume then. Notice is an even function so all are zero, and we can integrate over half the interval.

Let , . Then

Now let , . We have

Thus

Therefore

Matthew Leingang 2002-10-10