\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{color}
\usepackage{enumerate}
\usepackage{multicol}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\e{\mathbf{e}}
\def\v{\mathbf{v}}
\def\w{\mathbf{w}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\Per}{Per}
\DeclareMathOperator{\Poly}{Poly}
\DeclareMathOperator{\Span}{Span}


\def\<{\langle}
\def\>{\rangle}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}

\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due July 10 (Monday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2023 $|$ Dr.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: The Frobenius method and Bessel's equation}
\end{center}


\begin{enumerate}

\item The differential equation $(1-x^2)y''-xy'+p^2y=0$, where $p$ is
  a constant, is known as \emph{Chebyshev's equation}. It can be
  rewritten in the form
  \[
  y''+P(x)y'+Q(x)y=0\,,\qquad P(x)=-\frac{x}{1-x^2}\,,\quad
  Q(x)=\frac{p^2}{1-x^2}\,.
  \]
  \begin{enumerate}
  \item If $P(x)$ and $Q(x)$ are represented as a power series about
    $x_0=0$, what is the radius of convergence of these power series?
  \item Assume that the general solution has the form
    $\displaystyle\sum_{n=0}^\infty a_nx^n$, and find a recurrence for
    $a_{n+2}$ in terms of $a_n$. [\emph{Hint}: Before plugging back
    in, multiply through by $1-x^2$.]
  \item Use the recurrence to determine $a_n$ in terms of $a_0$ and
    $a_1$, for $2\leq n\leq 9$.
  \item For each $p\in\N$, there is a unique polynomial solution $T_p(x)$
    known as the \emph{Chebyshev polynomial} of degree $p$. Find $T_3(x)$. 
  \end{enumerate}
  


\item For each of the following ODEs, determine whether $x=0$ is an
  ordinary or singular point. If it is singular, determine whether it
  is regular or not. (Remember, first write each ODE in the form
  $y''+P(x)y'+Q(x)y=0$.)
  \begin{multicols}{2}
    \begin{enumerate}
    \item $y''+xy'+(1-x^2)y=0$
    \item $y''+(1/x)y'+(1-(1/x^2))y=0$.
    \item $x^2y''+2xy'+(\cos x)y=0$.
    \item $x^3y''+2xy'+(\cos x)y=0$.
    \end{enumerate}
  \end{multicols}

  

\item Consider the differential equation $3xy''+y'+y=0$. Since $x_0=0$
  is a regular singular point, there is a generalized power series
  solution of the form $\displaystyle y(x)=\sum_{n=0}^\infty
  a_nx^{n+r}$.
  \begin{enumerate}
  \item Determine the indicial equation (solve for $r$) and the
    recurrence relation for the coefficients.
  \item Find two linearly independent generalized power series
    solutions (i.e., a \emph{basis} for the solution space).
  \item Determine the radius of convergence of each of these
    solutions.  [\emph{Hint}: First compute the radius of convergence of
    $xP(x)$ and $x^2Q(x)$ and apply Frobenius].
  \end{enumerate}

  

\item Consider the differential equation $2xy''+y'+xy=0$. Since
  $x_0=0$ is a regular singular point, there is a solution of the form
  $\displaystyle y(x)=\sum_{n=0}^\infty a_nx^{n+r}$.
  \begin{enumerate}
  \item Determine the indicial equation (solve for $r$) and the
    recursion formula.
  \item Find a basis for the solution space and use this to write the
    general solution.
  \item What is the radius of convergence of each of these two linearly
    independent solutions?
  \end{enumerate}


\newpage

\item Consider the differential equation $xy''+2y'-xy=0$.
  \begin{enumerate}
  \item Show that $x=0$ is a regular singular point.
  \item Show that if $a_0=0$, then $r=-1$ is one solution for the
    indicial equation.
  \item For $r=-1$ and $a_0=0$, find the recurrence relation for
    $a_{n+2}$ in terms of $a_n$.
  \item Still assuming that $a_0=0$, write the solution from (b)
    as a generalized power series.
  \item Write this solution using a standard hyperbolic trigonometric
    function. \bigskip
  \end{enumerate}



% Asmar, Section 4.7, Exercise 19
\item Consider the ODE $y''+e^{-x}y=0$. Change variables by setting
  $t=2e^{-x/2}$, which will reduce this ODE to a Bessel's
  equation. Using the known solution to Bessel's equation,
  back-substitute to determine the solution $y(x)$ to the original
  ODE.


\end{enumerate}

\end{document}
