Math 6410 • ODE

Consider the problem
\[
\begin{aligned}
  Lu &\equiv -\frac{d}{dx}\left(p\frac{du}{dx}\right)+qu = \lambda u,\\
  \alpha u(0)+\beta\frac{du}{dx}(0) &= 0,\\
  \gamma u(1)+\delta\frac{du}{dx}(1) &= 0
\end{aligned}
\tag{5.1}
\]
where \(x\in[0,1]\), \(p\in C^1[0,1]\), \(q\in C[0,1]\), and
\(\alpha,\beta,\gamma,\delta\) are constants.
\begin{enumerate}
\item Show that (5.1) is self-adjoint if and only if \(p\) and \(q\) are real,
\(\gamma\bar{\delta}=\bar{\gamma}\delta\), and \(\alpha\bar{\beta}=\bar{\alpha}\beta\).
\item Suppose hereafter that \(p=q=1\), \(\alpha=1\), \(\beta=0\), \(\gamma=0\), \(\delta=1\).
Compute the eigenvalues and eigenfunctions for \(L\) with the given boundary conditions.
\item Compute the Green's function for \(L\) (with coefficients and boundary conditions
as in (b)).
\item Show how to solve the nonhomogeneous problem
\[
  Lu=f(x)
\]
with \(L\) and the boundary conditions as in (c). What conditions on \(f\) suffice to make
\(u\) a classical solution?
\end{enumerate}