Math 6420 • PDE

\begin{enumerate}
\item[(a)] Specify the weak formulation for the differential equation
\[
  u''+\lambda\delta(x-1/2)u=f(x)
\]
subject to boundary conditions \(u(0)=u(1)=0\).

\item[(b)] Find all values of \(\lambda\) for which this differential equation with
\(f(x)=0\) has a nontrivial solution and verify that the corresponding solution is a weak
solution.

\item[(c)] Suppose \(\lambda\) is such that this differential equation with \(f(x)=0\) has only
the trivial solution. Find the solution with \(f(x)=1\).
\end{enumerate}