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\title{Applied Analysis}
\author{Homework 6}
\date{due November 9, 2007}
\maketitle

\begin{enumerate}

\item Miller, p.\ 181, Exercise 5.18.

\item Consider the equation
\begin{gather*}
  y''(x) + (k^2 + \varepsilon \, \mu(x)) \, y(x) = 0 \,, \\
  y(0)=1 \,, \\
  y'(0) = 0 \,.
\end{gather*}
The first order regular perturbation approximation was computed in
class (or in Miller, pp.\ 258--260).
\begin{enumerate}
\item What is the interval of validity of the approximation which is
guaranteed by the abstract justification argument (as done in class or
in Miller, Section 7.1.3)?

\item Is this interval long or short as compared to the growth rate of
the secular term in the case of parametric resonance?

\item Investigate the full equation and its first order regular
perturbation approximation numerically (e.g.\ using
\emph{Mathematica}).  How should you quantify the accuracy of the
approximation?  What can you say about the interval of validity of the
asymptotic approximation?

\item Does the true solution exhibit secular growth beyond the
interval of validity of the first order approximation?  Can you
provide a rigorous analysis?

\end{enumerate}


\end{enumerate}
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