Math 6410 • ODE

Consider the wave equation
\[
  u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in\R.
\]
along with initial conditions
\[
  u(x,0)=g(x),\qquad u_t(x,0)=h(x).
\]
\begin{enumerate}
\item[(a)] Assuming \(c\) is constant, derive d'Alembert's Formula
\[
  u(x,t)=\frac{1}{2}[g(x+ct)+g(x-ct)]
  +\frac{1}{2c}\int_{x-ct}^{x+ct}h(\xi)\dd\xi.
\]
\item[(b)] Determine the solution for the initial data \(u(x,0)=1\) if \(\abs{x}<a\),
\(u(x,0)=0\) if \(x>\abs{a}\); \(u_t(x,0)=0\)
\item[(c)] Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if
\(\abs{x}<a\), \(u_t(x,0)=0\) if \(x>\abs{a}\)
\end{enumerate}