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Math 325


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\textbf{Take-home Problem for Final Exam}, 35 points
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Due at 4:15 p.m., Wednesday, May 9, 2001

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You may consult your course notes, homework, and the textbooks.
The problem requires using Maple.
You may not consult any other books or notes.  You may not discuss
the exam with anyone except Prof. Stanton.

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%The in-class part of the exam will be at 8:00 a.m. on May 8.  It will be
%closed book, but you may bring a sheet of paper ( $8\frac12 ''\times
%11''\/$) with notes 
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In this problem you will investigate the effect of a periodic, 
possibly
discontinuous,  forcing function  on a second order linear
equation with constant coefficients.  Consider the initial value
problem
\begin{eqnarray}
y''+y=h(t), \qquad y(0)=0, \ y'(0)=1.
\end{eqnarray}
The general solution of the  associated homogeneous equation is
$$
y(t)=A\cos t + B \sin t
$$
which is periodic with period  $2\pi\/$. 
Recall that the phenomenon of {\it resonance}, that is, solutions
which are unbounded as $t\to\infty$,  occurs when the forcing
function  $h(t)\/$ 
is a linear combination of  $\cos t\/$ 
and  $\sin t\/$.  (See p.\ 202 in Boyce and DiPrima for a discussion 
of resonance.)
You will investigate whether resonance occurs when the forcing function 
is periodic of period  $2\pi\/$  
but not necessarily continuous.
The problem is similar to Problem Set E \#17 in {\it Differential
Equations with Maple}.

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\emph{A note about terminology:} If you are asked to \textbf{explain} an
answer, you are being asked to give a mathematical justification, that
is, a mathematical proof that the answer is correct.

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(a) Using step functions, define a   function
 $H(t) \/$  a
whose value is  $t\/$ 
on $[0,2\pi)\/$, $t-2\pi\/$ on  $[2\pi,4\pi)\/$,  $t-4\pi\/$ on  
 $[4\pi,6\pi)\/$   
and so on.  
Define a Maple function  $h(t)\/$  which agrees with  $H(t)\/$
on the interval  $[0,10\pi]\/$. 
Plot the Maple function on the interval  $[0,30]\/$.  It should
have the appearance of a sawtooth wave.  
(Figure 6.3.8 on p. 316 of Boyce and DiPrima shows a sawtooth wave which
has period 1.) 

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(b) Use {\bf dsolve} with {\bf method=laplace}
to solve equation (1) with the  function  $h(t)\/$  defined in part (a). 
Explain why the solution will agree on the interval  $[0,30]\/$ 
with the one having  $H(t)\/$  forcing function.
 Plot the
solution together with  $h(t)\/$ 
on the interval  $[0,30]\/$.  Do you see resonance?  
Compute and plot a numerical solution to confirm your answer. 

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(c)  In part (a), you constructed a forcing function  $H(t)\/$ 
with period  $2\pi\/$.   
The function  $H(t/2)\/$  
has period  $4\pi\/$.  Repeat part (b) using the forcing function 
  $h(t/2)\/$.  Do you see resonance?   
If necessary, use a larger interval.

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(d) Repeat part (b) using the forcing function  $h(2t)\/$. 
%This time 
%just plot from  $0\/$ 
%to  $15\/$. 
 Do you see resonance?  What is the period of  $H(2t)\/$?

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(e)
What can you conclude about the resonance effect for      
 discontinuous forcing functions?  Would you expect resonance to occur
In equation (1) for {\it any} forcing function of period $2\pi$?       
 ({\it Hint}: Is the function $H(2t)$ periodic with period
$2\pi$?)  When do you expect resonance to occur for      
 piecewise continuous $2\pi\/$ periodic forcing functions? Explain your answer.
({\it Hint}:  Consider
the   $2\pi\/$ periodic 
Fourier series of the forcing function.  You might find it useful
to look at the section of problems on \emph{Periodic Forcing Terms} on 
p.\ 563 of Boyce and DiPrima.)  You
might try some other periodic forcing functions to check your answer.


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\textbf{Bonus question} (10 points):  
This winter and spring, the remarkable exhibit \emph{Witness \& Legacy: 
Contemporary Art About The Holocaust} has been at the Warner
Gallery of the South Bend Regional Museum of Art (at Century Center,
in downtown South Bend).  In conjunction with the exhibit, more than
twenty cultural institutions and colleges and universities in the
area,
including the University of Notre Dame, 
have sponsored events as part of \emph{In Unison: The 
Collaborative Community Programs of Witness and Legacy}.  Here is the 
description of the exhibit from the museum's web page 
(http://www.sbt.infi.net/$\widetilde{\ }$sbrma): 

\begin{itemize}
\item[{}] 
The South Bend Regional Museum of
Art (SBRMA) and the Kurt and Tessye Simon Fund for Holocaust
Remembrance are sponsoring an important and provocative exhibition
entitled Witness \& Legacy: Contemporary Art About The Holocaust.
Organized by the Minnesota Museum of American Art and the Regis
Foundation, this moving exhibition presents the work of twenty-two
American artists who include Holocaust survivors, "second generation"
children of survivors and others drawn to the subject matter by their
own humanitarianism and empathy. The artworks themselves incorporate a
diversity of forms in a variety of media including painting,
sculpture, photography, graphic design, and multimedia installation.
\end{itemize}

The purpose of this bonus question is to encourage you to see the
exhibit.
I recommend that you allow at least an hour.  I can't claim that 
the exhibit has anything to do with differential equations, but
there's at least a little related to mathematics in it. 
To get credit for the bonus question, you have to see the exhibit,
which is open Tuesday, Wednesday and Friday 11a.m.-5p.m., Thursday
11a.m.-9p.m.,
and  Saturday and Sunday noon-5p.m.,
and write a brief paragraph about it which mentions at least one
mathematical thing you found in the exhibit.  

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