Math 6410 • ODE

Consider the Sturm-Liouville operator
\[
\calL u=\frac{1}{w(x)}\{-(p(x)u')'+q(x)u\}
\]
where \(p(x)>0\), \(w(x)>0\) and \(1/p(x)\), \(q(x)\) and \(w(x)\) are locally Lebesgue
integrable on an arbitrary interval \(I\) (open or closed, bounded or unbounded).

\begin{enumerate}
\item[(a)] Define what is meant by a solution of \(\calL u=\lambda u\) on \(I\).

\item[(b)] Show that the initial value problem for the differential equation
\(\calL u=\lambda u\) on \(I\) with \(u(x_0)=c_1\), \((pu')(x_0)=c_2\) at \(x_0\in I\) is
equivalent to an integral equation for the vector
\((u_1,u_2)=(u,pu')\). Prove the equivalence.
\end{enumerate}