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\title{Midterm 1}
\date{}


\theoremstyle{definition} 
\newtheorem{definition}{Definition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{example}{Example}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{remark}{Remark}[section]
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\begin{document}
\maketitle
\begin{enumerate}
\item Solve the first order PDE
\begin{equation*}
\begin{cases}
-2u_x+u_y-yu=0 \\
u(x,0) = x^2.
\end{cases}
\end{equation*}
\item Attempt to find a particular solution of 
\begin{equation*}
3yu_x-2xu_y=0.
\end{equation*}  Determine if there is a unique solution, no solution, or infinitely many solutions for auxiliary conditions
\begin{enumerate}
\item $u(x,y) = x^2$ on the line $y=x$.
\item $u(x,y) = 1-x^2$ on the line $y=-x$.
\item $u(x,y) = 2x$ on the ellipse $2x^2+3y^2=4$.
\end{enumerate}

\item Show that the general solution of the wave equation $u_{tt}-c^2u_{xx}=0$ is given by 
\begin{equation*}
u(x,t) = f(x-ct) + g(x+ct)
\end{equation*}
for sufficiently differentiable functions $f$ and $g$ by considering the change of variables
\begin{align*}
s &= x-ct\\
r &= x+ct.
\end{align*}
\item Solve the initial value problem 
\begin{equation*}
\begin{cases}
u_{tt}-u_{xx}=0 & \text{ on } -\infty < x < \infty, t> 0\\
u(x,0) = 0\\
u_t(x,0) = \psi(x).
\end{cases}
\end{equation*}
where
\begin{equation*}
\psi(x)= \begin{cases}
1 & \text{ on } |x|\leq 1\\
0 & \text{ on } |x|>1.
\end{cases}
\end{equation*}
You may leave your solution in terms of $\psi$.  Using this solution, answer the following
\begin{enumerate}
\item Will the wave ever return to its original state at $x=0$?
\item What happens at each point $x$ when $t\to \infty$?
\item For some positive time $t>0$, is there a location $x$ such that $u(x,t) = 0$?
\end{enumerate}
\end{enumerate}


\end{document}


