Math 6410 • ODE

Consider the wave equation \(u_{tt}=u_{xx}\).

\begin{enumerate}
\item[(a)] State and derive the D'Alembert formula for the solutions \(u\) of this
equation.

\item[(b)] Use the method of separation of variables to solve the initial-boundary
value problem
\[
u_{tt}=u_{xx}\quad\text{for }0<x<1,\ t>0,
\]
\[
u(x,0)=f(x),\quad u_t(x,0)=0\quad\text{for }0\le x\le1,
\]
\[
u(0,t)=u(1,t)=0\quad\text{for }t\ge0.
\]
Define what you mean by a solution.

\item[(c)] Show how the solution of b) can be written in the D'Alembert form.
\end{enumerate}