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%  \def\emptyline{\vspace{12pt}}
% \DefineParaStyle{Maple Output}
% \DefineParaStyle{Maple Plot}
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\begin{document}

\textbf{Math 325: Differential Equations}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Final}\quad \textsl{May 9, 2001}

\bigskip







A table
of
Laplace transforms
and a table of Fourier series are attached.
  You may  use your own calculator.
You may also use a summary (two sides of an  $8\frac 12'' \times 11''\/$
sheet of paper, with notes in your writing).
You may not use anything else.  You may not pass a calculator
or summary
to another person.

\bigskip
Show all your work.  Erase or cross out any work you do not want
graded.
There are 6 questions on 8 pages plus the
bonus question page and two pages of tables.


\bigskip

1. (20 points)   Solve
$$
y'' + 16 y = 7\,\delta(t-\frac \pi 3),\qquad   y(0)=-2,\ y'(0)=1.
$$

\answerspace

\newpage
2.  (20 points) Let
$$
A := {\displaystyle \frac {1}{65}} \, \left[
{\begin{array}{rrrrr}
312 & 13 & 143 & 26 & -13 \\
-177 & 432 & 127 & 4 & 23 \\
398 & -3 & -23 & 74 & 3 \\
573 & 167 & 197 & 19 & -167 \\
215 & 90 & 170 & -205 & 365     
\end{array}}
 \right]
,\
B =
\left[
{\begin{array}{rrrrr}
1 & 1 & 0 & -1 & 0 \\
-1 & 2 & 1 & -1 & 0 \\
1 & 1 & 0 & 3 & -1 \\
1 & 2 & 0 & 1 & 4 \\
1 & 1 & 1 & 0 & 2
\end{array}}
 \right].
$$
Then
$$
B^{-1}AB=
\left[
{\begin{array}{rrrrr}
7 & 1 & 0 & 0 & 0 \\
0 & 7 & 0 & 0 & 0 \\
0 & 0 & 7 & 0 & 0 \\
0 & 0 & 0 & -2 & 1 \\
0 & 0 & 0 & 0 & -2
\end{array}}
 \right].
$$
Find the general solution of   ${\bf x}'=A{\bf x}\/$.


\newpage



3.  (20 points) Solve:
$$
{\bf x}''(t) +
\left[
{\begin{array}{rr}
-2 & 1 \\
1 & -2
\end{array}}
 \right]^2
{\bf x}(t)={\bf 0},\ {\bf x}(0)=\left[
{\begin{array}{r}
0 \\
0
\end{array}}
\right],
 \ {\bf x}'(0)=\left[
{\begin{array}{r}
12 \\
6
\end{array}}\right].
$$
({\bf Hint:} Suppose  $D\/$
is a diagonal matrix.  How would you solve  ${\bf y}''+D^2{\bf
y}=\bf 0\/$?)

\newpage

4. (15 points) Solve:
\begin{eqnarray*}
& & u_t =  u_{xx},
\qquad 0<x<\pi,\ t>0,
\vspace{1ex}\\
& & u(0,t) = 0=u(\pi,t),\qquad t>0,
\vspace{1ex}\\
& &u(x,0) = 73  \sin 5x -\sqrt{591} \sin 13 x ,\qquad 0<x<\pi.
\end{eqnarray*}

\newpage

5. (20 points)  The undamped pendulum satisfies the equations
\begin{eqnarray*}
& &x'=y
\vspace{1ex}
\\
& &
y'=-\sin x
\end{eqnarray*}
where  $x=\theta \/$
is the angle the pendulum makes with the downward vertical direction
and  $y=\theta '\/$
is the angular velocity.

\smallskip

(a) Find all equilibrium points of the system.

\answerspace


(b)  Show that the energy of the pendulum
$$
E(x,y)=\frac 12 y^2 + 1 -\cos x
$$
is constant on trajectories of the system.


\answerspace

(c)   Show that  $E(\pi,0)=2\/$
and  $E(0,2)=2\/$.

\answerspace


\newpage

(d)
Consider the following snapshot of Chuck's Maple session.

%% Created by Maple 6 (SUN SPARC SOLARIS)
%% Source Worksheet: pendulum.mws
%% Generated: Mon May  7 10:26:02 2001

 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Plot}
\DefineParaStyle{Warning}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(plots):}{%
}
\end{mapleinput}

\mapleresult
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{sys := diff(x(t),t) = y(t), diff(y(t),t)=-sin(x(t)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{numsol := b ->
dsolve(\{sys,x(0)=0,y(0)=b\},\{x(t),y(t)\},numeric):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{curve := (b,range) ->
odeplot(dsolve(\{sys,x(0)=0,y(0)=b\},\{x(t),y(t)\}          
,numeric),[x(t),y(t)],range,scaling=CONSTRAINED,numpoints=300):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{nphase := range ->
display(\{seq(curve(0.5*b,range),b=1..5)\},view=[-10..10,-5..5]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{nphase(0..20);}{%
}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{pendulum01.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}

%% End of Maple 6 Output


What is wrong with the plot?  What probably caused Maple to make
this mistake?

\newpage

6. (20 points)  Consider the following snapshot of Carol's Maple session.


%% Created by Maple 6 (SUN SPARC SOLARIS)
%% Source Worksheet: p6.mws
%% Generated: Mon May  7 10:12:52 2001

 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Plot}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{crit := solve(\{4*(y-x),-y*(2-x-y)*(-2-x-y)\},\{x,y\});}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{crit := \{y = 0, x = 0\}, \{y = -1, x = -1\}, \{y = 1, x = 1\};}{%
\[
\mathit{crit} := \{y=0, \,x=0\}, \,\{y=-1, \,x=-1\}, \,\{y=1, \,x
=1\}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(DEtools):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dfieldplot([diff(x(t),t)=4*(y(t)-x(t)),      
diff(y(t),t)=-y(t)*(2-x(t)-y(t))*(-2-x(t)-y(t))],
[x(t),y(t)],t=0..1,x=-2..2,y=-2..2,arrows=SLIM,axes=BOXED);}{%
}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{p601.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(linalg):}{%
}
\end{mapleinput}

\mapleresult
\begin{maplettyout}
Warning, the name adjoint has been redefined
\end{maplettyout}

\begin{maplettyout}
Warning, the protected names norm and trace have been redefined and
unprotected
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{jac := [[diff(4*(y-x),x),diff(4*(y-x),y)],
[diff(-y*(2-x-y)*(-2-x-y),x),diff(-y*(2-x-y)*(-2-x-y),y)]]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eigenvals(subs(crit[1],jac));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-4, 4;}{%
\[
-4, \,4
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eigenvals(subs(crit[2],jac));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-4+4*I, -4-4*I;}{%
\[
-4 + 4\,I, \,-4 - 4\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eigenvals(subs(crit[3],jac));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-4+4*I, -4-4*I;}{%
\[
-4 + 4\,I, \,-4 - 4\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}

%% End of Maple 6 Output


\newpage

(a) What system of differential equations is Carol studying?

\answerspace



(b)
What are the critical points of the system?


\answerspace



(c)
 Give its type and stability of each critical point.

\answerspace

 




(d)
For each critical point, give an initial condition 
\begin{math}
  (x_0,y_0) 
\end{math}
different from the critical point such that the corresponding solution
\begin{math}
  (x(t),y(t))
\end{math}
converges to this critical point.

\answerspace

\newpage



{\bf Bonus question } (10 points)

\noindent
Find:
$$
\sum_{n=1}^\infty \frac 1{n^2}.
$$

\newpage

\input table

\end{document}

