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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
$ \displaystyle\vphantom{\int}$ 1 15  
$ \displaystyle\vphantom{\int}$ 2 15  
$ \displaystyle\vphantom{\int}$ 3 15  
$ \displaystyle\vphantom{\int}$ 4 20  
$ \displaystyle\vphantom{\int}$ 5 15  
$ \displaystyle\vphantom{\int}$ 6 20  
$ \displaystyle\vphantom{\int}$ Total 100  

  1. (15 points)
    1. (10 points) Compute $ {\mathfrak{L}\left[1\right]}(s)$, using only the definition of Laplace transform.
    2. (5 points) Compute $ {\mathfrak{L}\left[t\right]}(s)$, using either the definition or by differentiating part (a).

  2. (15 points) Compute $ {\mathfrak{L}^{-1}\left[\frac{s}{(s-1)^{3/2}}\right]}$.

    Hint 1   $ s=s-1 + 1$.

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

    $\displaystyle y'' + 5y' + 6y = \delta(t),
\qquad y(0) = 3, \qquad y'(0) = 0.
$

  4. (20 points) Let $ a \in {\mathbb{R}}$ and suppose $ y$ is a solution to the ordinary differential equation

    $\displaystyle ty'' + y' + ta^2y = 0,
\qquad y(0) = 1, \qquad y'(0) = 0.
$

    1. (15 points) Show $ Y(s) = \frac{k}{\sqrt{s^2 + a^2}}$ for some constant $ k$.
    2. (5 points) Suppose $ \int_0^\infty y(t)\,dt = \alpha$. What is $ k$ in terms of $ \alpha$?

  5. (15 points) Let $ f(x) = \vert x\vert$ on $ [-\pi,\pi]$. Calculate the Fourier coefficients of $ f$ and write down the Fourier series.

  6. (20 points) Let $ 0 < L \leq \pi$ and define $ f(x) = \cos(x)$ on $ [-L,L]$. Calculate the Fourier coefficients of $ f$ and write down the Fourier series.



Matthew Leingang 2002-10-10