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\title{312 '10-EXAM 1}
\author{}
\maketitle
You may use one page of notes.  All problems are weighted equally.
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\begin{enumerate}
\item
Solve the PDE
\[ \frac{\partial u}{\partial x} + 2 \frac{\partial u}{\partial y} = e^{y - 2x} \ ,\ u(0,y) = 0 \ . \]

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\item
\begin{enumerate}
\item
Classify the following PDE's as elliptic, hyperbolic or parabolic.
\begin{eqnarray}
u_{xx} +   u_{xt} + 2 u_{tt} & = & 0 \\
u_{xx} + 2 u_{xt} +   u_{tt} & = & 0 \\
u_{xx} + 3 u_{xt} + 2 u_{tt} & = & 0 
\end{eqnarray}
\item
If any of the above PDE is hyperbolic, compute the solution to the initial value problem
\[ u(x,0) = \phi(x)\ ,\ u_t(x,0) = \psi(x)\ ,\  -\infty < x < \infty \ . \]
\end{enumerate}


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 \item
\begin{enumerate}
\item
Use separation of variables to solve the initial-boundary value problem:
\[ u_t = k u_{xx}\ ,\  u_x(0,t) = u_x(\pi,t) = 0\ ,\ u(x,0) = x\ ,\ 0\le x \le \pi \ . \]
\item What is the limit of $u(x,t)$ as $t \rightarrow \infty$?
\end{enumerate}

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\item
Consider the function
\[ f(x) = \left\{
\begin{array}{rr}
-1, & 0 < x < \frac{\pi}{2} \\
 1, & \frac{\pi}{2} < x < \pi \end{array} \right.  \ .\]
\begin{enumerate}
\item
Find the Fourier sine and cosine series expansion for $f$ over the interval $0\le x \le \pi$. 
\item
Give a sketch for the function represented by each over the interval $-2\pi \le x \le 2\pi$. 
\end{enumerate}

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