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 \centerline{\large Math.\ 401, Sec.\ 500
\hfill    Spring, 2005}
                                                                           
\bigskip
                                                                               

 
\centerline{\bfseries\large Homework 3, due February 9} 
\bigskip

\begin{enumerate}

 
 \item Find the first three terms (through order $\epsilon^2$) 
of 
the Taylor series (in $\epsilon$ with $t$ fixed) of
 $$ y(t)= {e^{-\epsilon t}\over \sqrt{1-\epsilon^2}}\,\sin 
(\sqrt{1-\epsilon^2}\,
t).$$
 Compare with the perturbative solution found in class (or 
notes) for the 
damped oscillator equation.
{\slshape Suggestion:\/}  The easiest method is
 to find series for each of the three
factors and then  multiply them together like polynomials.

%\item{2.} (dictionary research) \ Where did the phrase 
%``secular terms'' come from?  
%What does it have to do with the word ``secular'' in 
%``secular humanism''?

% \item{2-7.} Logan, p.\ 49, Ex.\ 1.3, 1.15, 1.16, 1.19.

\item 
{\itshape [Logan, p.\ 49,  Ex.\ 1.3, beginning]\/} \quad
Verify the following order relations:

\begin{enumerate}
\item $\epsilon^2 \tanh {\epsilon} = O(\epsilon^2)$ as 
$\epsilon\to 
\infty$.
\item $\exp (-\epsilon) = o(1)$ as $\epsilon\to \infty$.
\item $\sqrt{\epsilon(1-\epsilon)} = 
O\bigl(\sqrt{\epsilon}\bigr)$ as 
$\epsilon 
\to 0^+$.
\item ${\displaystyle{\sqrt{\epsilon} \over 1-\cos \epsilon}}
  = O(\epsilon^{-3/2})$ as $\epsilon \to 0^+$.
\end{enumerate}

\item 
{\itshape [Logan, p.\ 49,  Ex.\ 1.3, middle]\/} \quad
Verify the following order relations:

\begin{enumerate}
\item $\epsilon = O(\epsilon^2)$ as $\epsilon\to \infty$.

\item $\exp(\epsilon) -1 = O(\epsilon)$ as $\epsilon\to 0$.
\item $\int_0^\epsilon \exp(-x^2) \, dx = O(\epsilon)$ as 
$\epsilon 
\to 
0^+$.
\item $\exp(\tan \epsilon) = O(1)$ as $\epsilon\to 0$.
\end{enumerate}

\item 
{\itshape [Logan, p.\ 49,  Ex.\ 1.3, conclusion]\/} \quad
Verify the following order relations:


\begin{enumerate}
\item $e^{-\epsilon} = O(\epsilon^{-p})$ as $\epsilon\to \infty$, 
for all 
$p>0$.
\item $\ln{\epsilon} = o(\epsilon^{-p})$ as $\epsilon\to 0^+$, 
for all 
$p>0$.
 \end{enumerate}


 \item 
{\itshape [Logan, p.\ 49,  Ex.\ 1.16]\/}\quad
Show that the three-term expansion
$$\sin t + \epsilon \cos t - {\epsilon^2 \over 2}\,\sin t$$
is a uniformly valid approximation of $\,\sin (t+\epsilon)$
on $-\infty <t<\infty$.
 To justify your answer, look in a calculus book 
under  ``Taylor's theorem with remainder''\negthinspace. 
 
\vfill\eject

 \item 
{\itshape [Logan, p.\ 49,  Ex.\ 1.15]\/}\quad
Consider the boundary value problem (with $0<\epsilon \ll 1$)
$$Ly \equiv t^2 y'' + \epsilon t^2 y' + {\textstyle \frac14} y =0 
\quad 
\mbox{for $1\le t \le e$},$$
$$y(1) = 1, \qquad y(e) = 0.$$

\begin{enumerate}
\item Use regular perturbation theory to find the leading-order
behavior, $y_0(t)$.
{\slshape Hint:\/}
To solve the unperturbed equation,
 look in your differential equations textbook under ``Euler
equation'' ---
 or make a change of variable $t=e^u$.
\item Compute an upper bound for $|Ly_0|$ on $1\le t \le e$
when $\epsilon=0.01$.
\end{enumerate}

\item
{\itshape [Logan, p.\ 49,  Ex.\ 1.13]\/} \quad
The equation of motion of a pendulum can be scaled to the form
$$\frac {d^2\theta}{d\tau^2} + {\sin {A\theta}\over A} =0,$$
$$\theta(0)=1, \qquad \frac{d\theta}{d\tau}(0) = 0$$
(where $\tau>0$, $0<A\ll 1$).
Apply the regular perturbation method to find a two-term expansion.
Show that the correction term is secular and comment on the 
 validity of the approximation.

\end{enumerate}
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