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\begin{document}

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

\textbf{Final}\quad \textsl{May 9, 2000}

\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 9 pages plus the
bonus question page and two pages of tables.


\bigskip

1. (20 points)   Solve
$$
y'' + 4 y = 2\delta(t-\frac \pi 4),\qquad   y(0)=0,\ y'(0)=0.
$$

\answerspace

\newpage
2.  (20 points) Let
$$
A =\frac 1{65}
\, \left[
{\begin{array}{rrrrr}
190 & 10 & 120 & 35 & -10 \\
4 & 226 & 86 & 232 & -421 \\
385 & 10 & -75 & 35 & -10 \\
521 & 154 & 184 & -72 & -154 \\
366 & 9 & 134 & 38 & -204

\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}
5 & 1 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 & 0 \\
0 & 0 & -3 & 1 & 0 \\
0 & 0 & 0 & -3 & 0 \\
0 & 0 & 0 & 0 & -3
\end{array}}
 \right].
$$
Find the general solution of   ${\bf x}'=A{\bf x}\/$.


\newpage



3.  (20 points) Solve:
$$
{\bf x}''(t) +
\left[
{\begin{array}{rr}
-3 & 1 \\
1 & -3
\end{array}}
 \right]^2
{\bf x}(t)={\bf 0},\ {\bf x}(0)=\left[
{\begin{array}{r}
0 \\
0
\end{array}}
\right],
 \ {\bf x}'(0)=\left[
{\begin{array}{r}
8 \\
16
\end{array}}\right].
$$
({\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) = \sqrt {321} \sin 3x -59 \sin 7 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.

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

\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}


\begin{center}
%\mapleplot{pendulum01.eps}
{\includegraphics{pendulum01.eps}}
\end{center}

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

\end{maplegroup}

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.


\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{solve(\{-y*(x-3)=0,(x-2)*(x-2*y+1)=0\},\{x,y\});}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{\{x = 2, y = 0\}, \{y = 0, x = -1\}, \{y = 2, x = 3\};}{%
\[
\{x=2, \,y=0\}, \,\{y=0, \,x=-1\}, \,\{y=2, \,x=3\}
\]
%
}
\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)=-y(t)*(x(t)-3),diff(y(t),t)=(x(t)-2)*(x(t)-2
*y(t)+1) ],[x(t),y(t)],t=0..1,x=-2..4,y=-1..3, }{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{arrows=SLIM,axes=BOXED,scaling=CONSTRAINED);}{%
}
\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, new definition for adjoint
\end{maplettyout}

\begin{maplettyout}
Warning, new definition for norm
\end{maplettyout}

\begin{maplettyout}
Warning, new definition for trace
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{j1x := diff(-y*(x-3),x): j1y := diff(-y*(x-3),y):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{j2x := diff((x-2)*(x-2*y+1),x): j2y := diff((x-2)*(x-2*y+1),y):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{jacobmat := [[j1x,j1y],[j2x,j2y]]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{point1 := subs(\{x=3,y=2\},jacobmat):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{point2 := subs(\{x=-1,y=0\},jacobmat):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{point3 := subs(\{x=2,y=0\},jacobmat):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eigenvals(point1); eigenvals(point2); eigenvals(point3);  }{%
}
\end{mapleinput}

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

\begin{maplelatex}
\mapleinline{inert}{2d}{3+I*sqrt(3), 3-I*sqrt(3);}{%
\[
3 + I\,\sqrt{3}, \,3 - I\,\sqrt{3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{sqrt(3), -sqrt(3);}{%
\[
\sqrt{3}, \, - \sqrt{3}
\]
%
}
\end{maplelatex}

\end{maplegroup}

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

\end{maplegroup}


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

\answerspace


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


\answerspace

(c)
The Maple session contains enough information to identify the type and
stability of all but one of the critical points.  For each critical
point, give its type and stability or explain why the information in the
Maple session
is inconclusive.  Justify your answer based on the Jacobian information
found in the Maple session.

\answerspace

\ 

\answerspace

\newpage

(d)
If one imposes the initial condition  $x(0)=-1,\ y(0)=-0.1\/$,
the limiting behavior of the solution trajectory as $t\to\infty$
 is difficult to
ascertain.  Identify at least two possibilities for the limiting behavior
of the solution trajectory.


\newpage



{\bf Bonus question } (10 points)

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

\newpage

\input table

\end{document}

