Math 6410 • ODE

Consider the problem
\[
\begin{aligned}
  u_{tt}-u_{xx} &= 0,\\
  u(x,0) &= F(x),\\
  u_t(x,0) &= 0,\\
  u(0,t)=u(\pi,t) &= 0
\end{aligned}
\tag{7.1}
\]
\begin{enumerate}
\item Use separation of variables to formally solve (7.1) when \(F\in C[0,\pi]\).
\item Suppose that \(F\in C^4[0,\pi]\) and that \(F''(0)=F''(\pi)=0\). Prove that the
series you obtain in (a) defines a classical solution of (7.1) under the foregoing
conditions on \(F\).
\end{enumerate}