Math 6410 • ODE

Consider the Sturm-Liouville problem
\[
  x^2(x^2u')'+\lambda u=0
\]
on \(1/2\le x\le1\) with boundary conditions \(u(1/2)=u(1)=0\).
\begin{enumerate}
\item[(a)] State without proof the main theorem on eigenfunction expansions which
applies to this example.
\item[(b)] Use the change of variables \(w(t)=u(1/t)\) to transform the differential
equation into \(d^2w/dt^2+\lambda w=0\). Then calculate the eigenvalues \(\lambda_n\) and
eigenfunctions \(u_n\), by citing without proof a result for the Sturm-Liouville problem
\(y''+\lambda y=0\), \(y(a)=0=y(b)\).
\item[(c)] Sturm oscillation theory and the Pr\"ufer transformation are used in the
general theory to produce the candidate eigenvalues and eigenfunctions. Sketch briefly
how this is accomplished, without proofs.
\end{enumerate}