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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due April 3
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent For consistency, we will say that a \emph{Sturm-Liouville
  equation} is a second-order differential equation in the following
\emph{self-adjoint form}:
\begin{equation}
  \label{eq:self-adjoint}
  -\frac{d}{dx}(p(x)y')+q(x)y=\lambda w(x)y\,,
\end{equation}
where $p(x)>0$ and $w(x)>0$ is called the \emph{weight}, or
\emph{density} function. If we divide through by $w(x)$, we can write this
equation as $Ly=\lambda y$, where $L$ is a \emph{self-adjoint} linear
operator. The possible values of $\lambda$ are the \emph{eigenvalues},
and solutions are the \emph{eigenfunctions}.


\begin{enumerate}
  
\item Write the following differential equations in self-adjoint
  form. That is, put them in the above form, and find $p(x)$, $q(x)$,
  and the \emph{weight} $w(x)$. Also, write out the corresponding
  linear operator $L$.
  \begin{enumerate}
  \item $-y''=\lambda y$
  \item Bessel's equation: $-xy''-y'=\lambda xy$
  \item Airy's equation: $y''+(\lambda-x)y=0$
  \item Hermite's equation: $y''-2xy'+\lambda y=0$ [\emph{Hint}:
    First, multiply through by an \emph{integrating factor}, $e^{\int
      \!-2x\,dx}$.]
  \item Chebyshev's equation: $(1-x^2)y''-xy'+\lambda y=0$
    [\emph{Hint}: Divide through by $\sqrt{1-x^2}$ first.]
%  \item An arbitrary linear equation: $y''+P(x)y'+Q(x)y=\lambda R(x)y$
%    [\emph{Hint}: Multiply through by an \emph{integrating factor},
%      $e^{\int P(x)dx}$.]
%  \item The equation $x^3y''-xy'+2y=\lambda y$. 
  \end{enumerate}
  
  
\item In this problem, we will find all solutions to the
  Sturm-Liouville problem $-y''=\lambda y$, $y'(0)=y'(L)=0$, where
  $\lambda$ is a constant.
  \begin{enumerate}
  \item First, suppose that $\lambda=0$. That is, solve $y''=0$,
    $y'(0)=y'(L)=0$.
  \item Next, suppose $\lambda=-\omega^2\leq 0$.  That is, solve the
    boundary value problem $y''=\omega^2 y$,
    $y'(0)=y'(L)=0$. [\emph{Hint}: When the domain is finite, e.g.,
      $[0,L]$, it is usually more convenient to use $\cosh$ and
      $\sinh$ instead of exponentials.]
  \item Finally, suppose $\lambda=\omega^2>0$. That is, solve
    $y''=-\omega^2 y$, $y'(0)=y'(L)=0$.
  \item Summarize the results from parts (a)--(c) in terms of the
    \emph{eigenvalues} $\{\lambda_n\mid n=0,1,2,\dots\}$ and
    corresponding \emph{eigenfunctions} $\{y_n(x)\mid n=0,1,2,\dots\}$
    of a particular linear differential operator $L$. What is $L$?
  \end{enumerate}


\item In this problem, we will consider a simple differential equation
  under several different types of boundary conditions.
  \begin{enumerate}
  \item Find the eigenvalues and eigenfunctions of the following
    Sturm-Liouville problem under \emph{Dirichlet} boundary
    conditions:
    \[
    -y''=\lambda y\,,\qquad y(0)=0\,,\quad y(\pi)=0\,.
    \]
    Additionally, sketch $y_1(x)$, $y_2(x)$, and $y_3(x)$ on
    $[0,\pi]$.
  \item Find the eigenvalues and eigenfunctions of the following
    Sturm-Liouville problem under \emph{Neumann} boundary
    conditions:
    \[
    -y''=\lambda y\,,\qquad y'(0)=0\,,\quad y'(\pi)=0\,.
    \]
    Additionally, sketch $y_0(x)$, $y_1(x)$, $y_2(x)$, and $y_3(x)$ on
    $[0,\pi]$.
  \item Find the eigenvalues and eigenfunctions of the following
    Sturm-Liouville problem under \emph{mixed} boundary
    conditions:
    \[
    -y''=\lambda y\,,\qquad y(0)=0\,,\quad y'(\pi)=0\,.
    \]  
    Additionally, sketch $y_1(x)$, $y_2(x)$, and $y_3(x)$ on
    $[0,\pi]$.
  \end{enumerate}



\item By the main theorem of Sturm-Liouville theory, if we define an
  inner product as
  \begin{equation}
    \label{eq:SL-innerproduct}
    \<f,g\>=\int_a^b f(x)\overline{g(x)}w(x)\,dx\,, 
  \end{equation}
  then the eigenfunctions $\{y_n(x)\}$ form an \emph{orthogonal basis}
  (Note: not necessarily \emph{orthonormal}!) for the space of
  functions, integrable on $[a,b]$ with $\<f,f\><\infty$ that satisfy
  the boundary condtions. This means that for any $f\in L^2([a,b],w)$
  with the same boundary conditions, we can write
  \[
  f(x)=\sum_{n=1}^\infty c_ny_n(x)\,.
  \]
  \begin{enumerate}
  \item Consider the Sturm-Liouville problem from Part~(c) of the
    previous problem:
    \[
    -y''=\lambda y\,,\qquad y(0)=0\,,\quad y'(\pi)=0\,.
    \]
    What is $w(x)$?
  \item The function $f(x)=x(2\pi-x)$ is clearly continuous and
    satisfies $f(0)=f'(\pi)=0$. Compute the \emph{norm} of $f$. Recall
    that this is defined as $||f||=\<f,f\>^{1/2}$.
  \item Since the eigenfunctions form a basis for the subspace of
    $L^2([0,\pi];1)$ that satisfy the above boundary conditions, we
    can write
    \[
    x(2\pi-x)=\sum_{n=1}^\infty c_ny_n(x)\,,\qquad 0\leq x\leq\pi\,.
    \]
    Write down a formula for the $c_n$'s. Leave your answer in terms
    of an integral -- no need to actually compute it!  [\emph{Hint}:
      Don't forget that $y_n(x)$ isn't necessarily of unit length!]
  \end{enumerate}


\item Consider the following Sturm-Liouville problem:
  \[
  -y''-y'=\lambda y\,,\qquad y(0)=0\,\quad y(2)=0\,.
  \]
  \begin{enumerate}
  \item Find the eigenvalues and eigenfunctions. [\emph{Hint}: You
    will encounter a \emph{discriminant} of $D=1-4\lambda$. As before,
    there will be three cases: $D=0$, $D>0$, and $D<0$.]
  \item Write this differential equation in standard form, as in
    Equation~\eqref{eq:self-adjoint}. [\emph{Hint}: First, multiply through
      by an \emph{integrating factor}, $e^x$.]
  \item Write a formula for $\<y_n(x),y_m(x)\>$ in terms of an
    integral. What is this integral equal to when $n\neq m$?
  \end{enumerate}


\newpage

\item Consider the following Sturm-Liouville equation on $[-1,1]$, called
  \emph{Legendre's differential equation}:
  \begin{equation}
    \label{eq:legendre}
    (1-x^2)y''-2xy'+\lambda y=0
  \end{equation}
  In this problem, you will find the eigenvalue and eigenfunctions,
  which have already come up several times in this class in different
  settings.
  \begin{enumerate}
  \item Write Legendre's equation into \emph{self-adjoint form}, as in
    Equation~\ref{eq:self-adjoint}. That is, find $p(x)$, $q(x)$, and
    $w(x)$, and the self-adjoint operator $L$. This is called a
    \emph{singular Sturm-Liouville problem} on the inteveral
    $[a,b]=[-1,1]$ because the function $p(x)$ satisfies
    $p(-1)=p(1)=0$, and so boundary conditions on $y(x)$ are not
    needed.
  \item Assume that there is a power series solution of the form
    $\displaystyle\sum_{n=0}^\infty a_nx^n$. Plug your power series
    solution into Equation~\ref{eq:legendre} and find the recurrence
    relation for the coefficients.
  \item Recall from HW 4 that a general power series solution will
    have radius of convergence $R=1$, i.e., it will be defined on the
    open interval $(-1,1)$, but \emph{not} on its endpoints, $a=-1$ or
    $b=1$. However, if we have a \emph{polynomial} solution (that is,
    only finitely many non-zero terms, which happens when $a_{n+2}=0$
    for some $n$), then this will certainly be defined on all of
    $[-1,1]$. What values of $\lambda$ lead to a polynomial solution?
    (These are the \emph{eigenvalues} of $L$.)
  \item The \emph{eigenfunction} for eigenvalue $\lambda_k$ is a
    polynomial $P_k(x)$ called the \emph{Legendre polynomial} of
    degree $k$. (These arose on HW 4 and HW 6.) By Sturm-Liouville
    theory, they form an \emph{orthogonal basis} of $L^2([-1,1])$,
    meaning that
    \[
    \<P_n(x),P_m(x)\>:=\int_{-1}^1 P_n(x)P_m(x)\,dx=0\,,\qquad n\neq m\,.
    \]
    Use the recurrence relation to write out the first five Legendre
    polynomials, $P_k(x)$, for $k=0,\dots,4$. Normalize each one so
    they form an \emph{orthonormal} set. 
  \item Write the polynomial $f(x)=3x^3-2x^2+4$ using the first four
    Legendre polynomials. That is, find $C_0$, $C_1$, $C_2$, and $C_3$
    such that
    \[
    3x^3-2x^2+4=C_0P_0(x)+C_1P_1(x)+C_2P_2(x)+C_3P_3(x)\,.
    \]
    \emph{Hint}: This is \emph{very} similar to a problem you did on HW 6!
  \end{enumerate}


\end{enumerate}

\end{document}






% Part of lecture instead
\item Consider a Sturm-Liouville equation $Ly=\lambda y$, where $L$ is
  the linear differential operator as in
  Equation~\eqref{eq:self-adjoint}, and the corresponding inner product
  as in Equation~\eqref{eq:SL-innerproduct}. The purpose of this problem is to
  understand why the eigenfunctions form an orthogonal set.
  \begin{enumerate}
  \item One reason why a Sturm-Liouville equation is typically written
    in so-called ``self-adjoint'' form as in
    Equation~\ref{eq:self-adjoint} is because the corresponding linear
    operator $L$ is \emph{self-adjoint}, meaining that the following
    relation holds (this is not obvious):
    \[
    \<L(f),g\>=\<f,L(g)\>.
    \]
    This property of linear operators is analogous to a matrix (a
    linear map) being symmetric. Write out formulas for $\<L(f),g\>$
    and $\<f,L(g)\>$, using the definition of the inner product.  It
    can be verified that these are equal using integration by parts,
    but you do not need to do this. (It's messy.)
  \item Suppose that $y_i(x)$ and $y_j(x)$ are eigenfunctions
    corresponding to distinct eigenvalues
    $\lambda_i\neq\lambda_j$. Use the equality above to show that
    $\<y_i(x),y_j(x)\>=0$, i.e., that $y_i(x)$ and $y_j(x)$ are
    orthogonal. [\emph{Hint}: Let $f(x)=y_i(x)$ and $g(x)=y_j(x)$.]
  \end{enumerate}
