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\def\Rl{\mathbb{R}}
\def\Cx{\mathbb{C}}

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\centerline{{\sc Problem Set 3}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Feb. 3}


\bigskip\noindent
{\bf 1.} Suppose that $u(x)$ is a solution of the
Sturm-Liouville problem with non-homogeneous ODE and BCs
\begin{align*}
&-(pu')' + q u = f(x) \qquad a<x<b,
\\
&u(a) = A,\qquad u(b) =B.
\end{align*}
Write
\[
u(x) = A\left(\frac{b-x}{b-a}\right) + B \left(\frac{x-a}{b-a}\right) + v(x)
\]
and show that $v$ satisfies a Sturm-Liouville problem of the form
\begin{align*}
&-(pv')' + q v = g(x) \qquad a<x<b,
\\
&v(a) = 0,\qquad v(b) =0
\end{align*}
with homogeneous BCs.

\bigskip\noindent
{\bf 2.} Consider the nonhomogeneous Sturm-Liouville problem
\begin{align*}
&-(pu')' + q u = \lambda u + f(x) \qquad a<x<b,
\\
&u(a) = 0,\qquad u(b) =0.
\end{align*}
If $\lambda$ is an eigenvalue with eigenfunction $\phi$, show that the problem only has a solution if $f$ satisfies
\[
\int_a^b f \overline{\phi}\, dx = 0.
\]
Under what conditions on $f$ is the BVP
\begin{align*}
&-u'' = f(x) \qquad 0<x<1,
\\
&u'(0) = 0,\qquad u'(1) =0
\end{align*}
solvable? How about the BVP
\begin{align*}
&-u'' = f(x) \qquad 0<x<1,
\\
&u'(0) = 0,\qquad u'(1) =1.
\end{align*}



\newpage\noindent
{\bf 3.} Consider the weighted Sturm-Liouville eigenvalue problem
\begin{align*}
&-(pu')' + q u = \lambda r u \qquad a<x<b,
\\
&u(a) = 0,\qquad u(b) =0
\end{align*}
where $p(x)$, $q(x)$, $r(x)$ are given real-valued
coefficient functions and $r >0$. Let $L^2_r(a,b)$ denote the space of functions
$f:[a,b]\to \Cx$ such that
\[
\int_a^b r |f|^2\,dx < \infty
\]
with weighted inner product
\[
(f,g)_r = \int_a^b r f\overline{g}\, dx.
\]
(a) If $\phi(x)$ is an eigenfunction with eigenvalue $\lambda\in \Cx$, show that $\lambda \in \Rl$ is real.

\smallskip\noindent
(b) If $\phi(x)$, $\psi(x)$ are eigenfunctions with distinct eigenvalues $\lambda$, $\mu$ show that
they are orthogonal with respect to the weighted inner-product, meaning that
\[
\int_a^b r \phi \overline{\psi}\, dx = 0.
\]

\smallskip\noindent
(c) Suppose that the eigenvalue problem has a complete set of eigenfunctions $\{\phi_n : n= 1, 2, 3,\dots\}$. If $f\in L^2_r(a,b)$, give
an expression for the coefficients $c_n$ in the eigenfunction expansion
\[
f(x) = \sum_{n=1}^\infty c_n \phi_n(x).
\]


\bigskip\noindent
{\bf 4.} Use separation of variables to solve the following IBVP for $u(x,t)$
for the wave equation:
\begin{align*}
&u_{tt} = u_{xx} \qquad\qquad 0<x<1,
\\
&u_x(0,t) = 0,\qquad u(1,t) = 0,
\\
&u(x,0) = f(x),\qquad u_t(x,0) = g(x).
\end{align*}
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