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 \centerline{\large Math.\ 401, Sec.\ 501
\hfill    Spring, 2006}
                                                                            
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\centerline{\bfseries\large Homework 4, due February 15}
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\begin{enumerate}

\item
{\itshape [Logan, p.\ 49,  Ex.\ 1.19]\/}\quad
Apply the Poincar\'e--Lindstedt method to the scaled pendulum
problem (stated in the last exercise of Homework~3).
Find a two-term perturbation solution.

 \item By the Poincar\'e  (distorted-time) method,
 find a good one-term approximation to the solutions of 
$$\frac{d^2y}{dt^2} + [\omega^2 -V(t)]y =0 \eqno(1) $$
 when $\omega\to+\infty$ (i.e., $\epsilon=\omega^{-1}$ is the 
small parameter), 
and 
\begin{enumerate}
\item \ $\displaystyle V(t) =\frac1{1+t^2}\,.$ 
 \item \ $V$ is an  arbitrary continuous function.
\end{enumerate}
 In both cases, assume that $\omega^2 -V(t)$ is positive in the 
interval of interest.  (See Problem~4 below for more about that.) 

   \item
\begin{enumerate}
\item Apply the Poincar\'e  method to
 $$\frac{d^2y}{dt^2} + \epsilon \left(\frac{dy}{dt}\right)^3 + y 
=0,
 \qquad y(0) =1, \quad \frac{dy}{dt}(0) =0.$$
 Carry the calculations far enough to demonstrate that that method
{\itshape doesn't work\/}  on this problem.
 \item  Treat the problem by the method of two time scales.
 {\slshape Hint for solving the two coupled nonlinear 
equations:\/}
 Try setting one function identically equal to~0.
  \item Why did method (a) fail and method (b) work?
 [What makes this equation different from, say,
 $y'' +y +\epsilon y^3 =0\,$?]
\end{enumerate}


  \item In our study of the equation
  $\displaystyle\frac{d^2y}{dt^2} + [\omega^2 -V(t)]y =0  $
 we assumed that $\omega^2 -V(t)$ is positive.
 Give a critical discussion of the following claim:
$$\hbox{This condition will always be satisfied if $\omega$ is 
sufficiently large.}$$
%
%\zitem{}This condition will always be satisfied if \zw (assumed real and
%positive) is sufficiently large.
%
%
%\medskip
%{\narrower\narrower  This condition will always be 
%satisfied if 
%$\omega$ is sufficiently large. \par}
%\medskip
%
%\noindent 
Is it true, false, or a half-truth?  {\slshape 
Hint:\/} 
Remember the
 distinction between pointwise and uniform limits.
(It is understood that $\omega$ is real.)

\end{enumerate}
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