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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 27
    (Monday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


\item Consider the following initial value problem:
  \[
  y'''-y=0\,,\qquad y(0)=1\,,\qquad y'(0)=0\,,\qquad y''(0)=0\,.
  \]
  \begin{enumerate}
  \item Assume there is a solution of the form $y(t)=e^{rt}$, and
    plug this back in and solve for $r$. Use this to write the general
    solution. [\emph{Hint}: The equation $r^3=1$ has 3 distinct
      solutions over $\C$, called the \emph{3rd roots of
        unity}: $r_1=e^{0\pi i/3}=1$, $r_2=e^{2\pi i/3}$, and
      $r_2=e^{4\pi i/3}$.]
  \item From here, you could solve the initial value problem by
    plugging in the initial conditions, but you'll get a system of
    three equations with three unknowns, and involving complex
    numbers. Here's another way: look for a power series solution,
    $\displaystyle y(t)=\sum_{n=0}^\infty a_nt^n$. Plug this back into
    the ODE and find the recurrence relation for the coefficients.
  \item Compute $a_n$ for $n\leq 10$, and once you see the pattern,
    write down the general solution to the ODE as a power
    series. [\emph{Hint}: It should look familiar!] 
  \item Write down a \emph{basis} for the solution space. 
  \item Plug in the initial conditions and find the particular
    solution to the initial value problem.
  \item Consider a similar initial value problem:
    \[
    y'''-y=0\,,\qquad y(0)=1\,,\qquad y'(0)=1\,,\qquad y''(0)=1\,.
    \]
    Find $y(t)$.
\end{enumerate}


\item Find the radius of convergence of the following power series.
  \[
  \begin{array}{lllll}
    \displaystyle(\mbox{a})\quad\sum_{n=0}^\infty(-1)^nt^n &&
    \displaystyle(\mbox{b})\quad\sum_{n=0}^\infty\frac{1}{n+1}(t-\pi)^n && 
    \displaystyle(\mbox{c})\quad\sum_{n=0}^\infty\frac{3}{n+1}(t-2)^n, \\
    \displaystyle(\mbox{d})\quad\sum_{n=0}^\infty\frac{1}{2^n}(t-\pi)^n && 
    \displaystyle(\mbox{e})\quad\sum_{n=0}^\infty(5t-10)^n &&
    \displaystyle(\mbox{f})\quad\sum_{n=0}^\infty\frac{1}{n!}(3t-6)^n
  \end{array}
  \]


\item The differential equation $(1-t^2)y''-ty'+p^2y=0$, where $p$ is
  a constant, is known as \emph{Chebyshev's equation}. It can be
  rewritten in the form
  \[
  y''+P(t)y'+Q(t)y=0\,,\qquad P(t)=-\frac{t}{1-t^2}\,,\quad
  Q(t)=\frac{p^2}{1-t^2}\,.
  \]
  \begin{enumerate}
  \item If $P(t)$ and $Q(t)$ are represented as a power series about
    $t_0=0$, what is the radius of convergence of these power series?
  \item Assume that the general solution has the form
    $\displaystyle\sum_{n=0}^\infty a_nt^n$, and find a recurrence for
    $a_{n+2}$ in terms of $a_n$. [\emph{Hint}: Before plugging back
    in, multiply through by $1-t^2$.]
  \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 $T_p(t)$
    known as the \emph{Chebyshev polynomial} of degree $p$. Find $T_3(t)$. 
  \end{enumerate}
  

\item For each of the following ODEs, determine whether $t=0$ 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
  $y''+P(t)y'+Q(t)y=0$.)
  \begin{enumerate}
  \item $y''+ty'+(1-t^2)y=0$
  \item $y''+(1/t)y'+(1-(1/t^2))y=0$.
  \item $t^2y''+2ty'+(\cos t)y=0$.
  \item $t^3y''+2ty'+(\cos t)y=0$.
  \end{enumerate}
  

\item Consider the differential equation $3ty''+y'+y=0$. Since $x_0=0$
  is a regular singular point, there is a generalized power series
  solution of the form $\displaystyle y(t)=\sum_{n=0}^\infty
  a_nt^{n+r}$.
  \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
    $tP(t)$ and $t^2Q(t)$ and apply Frobenius].
  \end{enumerate}
  

\item Consider the differential equation $2ty''+y'+ty=0$. Since
  $t_0=0$ is a regular singular point, there is a solution of the form
  $\displaystyle y(t)=\sum_{n=0}^\infty a_nt^{n+r}$.
  \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}


\item Consider the differential equation $ty''+2y'-ty=0$.
  \begin{enumerate}
  \item Show that $t=0$ 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.
  \end{enumerate}

  
\end{enumerate}

\end{document}
