Math 6410 • ODE

\begin{enumerate}

\item
What are necessary conditions on the parameters \(a\) and \(b\) so that the boundary value
problem

\[
u''+u=\cos(x),\quad u'(0)=a,\quad u'(\pi)=b
\]

has a solution?


\item
What are the eigenvalues and eigenfunctions for the operator \(Lu=u''+u\) with
homogeneous boundary conditions \(u'(0)=0\), \(u'(\pi)=0\)? Show that the
eigenfunctions for this operator are the same as those for \(Lu=u''-u\) with
homogeneous boundary conditions \(u'(0)=0\), \(u'(\pi)=0\). Outline the proof
that these eigenfunctions form a
complete set. You may invoke a theorem associated with symmetric, compact, linear
operators. Be sure to specify the function space on which this result applies.

\end{enumerate}