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\textbf{Math 325: Differential Equations}
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\textbf{Quiz 10}\quad \textsl{April 20, 2001}
 
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You may  use your own calculator.
You may not use anything else.  You may not pass a calculator 
to another person.

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Show all your work.  Erase or cross out any work you do not want graded.

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Find the solution of  the heat equation
$$
u_{xx}(x,t)=u_t(x,t), \qquad  0<x<1, \ 0<t,
$$
satisfying the initital conditon
$$
u(x,0)= 351\sin 5\pi x, \qquad 0<x<1,
$$
and the boundary conditions
$$
u(0,t)=0=u(1,t),\qquad t>0.
$$

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{\bf Bonus Questions} (5 points each) These questions can be done very
quickly and in a short amount of space.  I will not read lengthy
solutions.

a) Find the Fourier series of 
\begin{math}
  \cos(4235679801 x)
\end{math}
 on the interval  
 \begin{math}
   -\pi < x < \pi.
 \end{math}


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b) Find the Fourier series of 
\begin{math}
  \cos^2(\pi x)
\end{math}
on the interval 
\begin{math}
  -1<x<1.
\end{math}

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