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\centerline{Mathematics 304, Section 2 -- Ordinary Differential Equations}
\centerline{Final Examination -- December 16, 2004}
\bigskip
\bigskip
\noindent
{\it Directions:}
\bigskip
Do all work in the blue exam booklet.  You may use a calculator and
the table of integrals at any point.  There are 200 total regular
points and 20 possible Extra Credit points.
\vskip 20pt
\noindent
I.  Consider the first order ODE 
$$y' = \cos(t)(y-2)^{\frac{1}{3}}\leqno(1)$$
\begin{itemize}
\item[A)] (15)  Find a solution of the initial value problem 
for (1) with $y(0) = 3$. 
\item[B)]  (15) What does the Existence and Uniqueness Theorem say
about solutions of the initial value problem for (1) with
$y(0) = 2$?  Show that the constant function $y(t) \equiv 2$ is 
one solution.  Is it the only one?
\end{itemize}

\bigskip
\noindent
II.  (25)  Sketch the bifurcation diagram for the family
of 1st order ODE given by $x' = x^2 - a$, where $a \in {\bf R}$
is an arbitrary real parameter.  In your sketch, show the equilibrium
points for the different equations in the family, and identify 
their types. 

\bigskip
\noindent
III.  The temperature $y$ of an insulated shed with no internal 
heating or cooling varies according to the following ODE
(from Newton's Law of Cooling):
$$y' = (1.5)(A(t) - y).\leqno(2)$$
In this equation, $A(t)$ is the outside temperature, which varies as
a scaled and shifted cosine function with a minimum of 
$40^\circ$ F at 12 midnight and a maximum of $70^\circ$ F at 12 noon.  
\begin{itemize}
\item[A)] (5) Letting $t = 0$ correspond to 12 noon, write
a formula for $A(t)$.
\item[B)] (15) Using your answer from part A, solve (2) for $y(t)$. 
\item[C)] (5) Your answer from part B should have an arbitrary
constant.  Does the value of that constant in a particular
solution affect the {\it long-term behavior} of $y$?  Explain.
\end{itemize}

\bigskip
\noindent
IV.  Consider the family of first order systems:
$$X' = \begin{pmatrix} a & 4a \\ 
                       1 & 0\\ 
       \end{pmatrix}X,$$
where $a$ is a real parameter.
\begin{itemize}
\item[A)]  (5)  For what range of $a$-values will this
system have a {\it spiral sink} at $(0,0)$?
\item[B)] (15)  Find the general solution and sketch the phase portrait
of the system in this family with $a = 2$ .
\end{itemize}

\bigskip
\noindent
V.  (20)  Determine the general solution of the first
order system 
$$X' = \begin{pmatrix} 2 & 0 & 1\\
                       1 & 2 & 0\\
                       0 & 0 & -3\\
       \end{pmatrix} X$$
using the canonical form of the coefficient matrix
and change of basis.


\bigskip
\noindent
VI.  Consider the following 2nd order ODE.
$$y'' + 6 y' + 9y = 3e^{-t} + t^2\leqno(3)$$
\begin{itemize}
\item[A)] (10) Find the general solution of the associated
homogeneous equation.
\item[B)] (10) Find a particular solution of (3) by the method 
of undetermined coefficients.
\item[C)] (10) How would the form of the solutions of (3)
change if the coefficient of the $y$ term was increased
to $9 + \varepsilon$ for $\varepsilon > 0$? 
\end{itemize}

\bigskip
\noindent
VII.  All parts of the following problem refer to the 
1st order system
\begin{eqnarray*} x' &=& -y + x + 1\\
                  y' &=& x(x - 2y + 1).\\
\end{eqnarray*}
\begin{itemize}
\item[A)] (10) Find all the equilibrium points.
\item[B)] (15) Compute the linearization at each
critical point and use that information to determine the type of each 
(sink, source, saddle, or center).  
\end{itemize}

\bigskip
\noindent
VIII.
\begin{itemize}
\item[A)] (5) Is the following first order system Hamiltonian?  Why or why not?
\begin{eqnarray*} x' &=& x e^{xy}\\
                  y' &=& -y e^{xy}\\
\end{eqnarray*}
\item[B)] (10) Show that the linearization of a Hamiltonian
system at an equilibrium always has the form $V' = AV$, 
where ${\rm Tr}(A) = 0$.
\item[C)]  (10) Which of the following phase portraits could
show Hamiltonian systems? (There could be several.)
\vfill\eject
\centerline{\psfig{figure=1.ps,height=2.0in,width=2.4in,angle=-90}
$\qquad$
\psfig{figure=2.ps,height=2.0in,width=2.4in,angle=-90}}
\centerline{\psfig{figure=3.ps,height=2.0in,width=2.4in,angle=-90}}
\end{itemize}
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\noindent
{\it Extra Credit.} A non-linear system 
\begin{eqnarray*}
x' &=& f(x,y) \\
y' &=& g(x,y) \\
\end{eqnarray*}
is said to be a {\it gradient system} on $U \subset {\bf R}^2$
if there is some function 
$G(x,y)$ such that $f(x,y) = \frac{\partial G}{\partial x}$
and $g(x,y) = \frac{\partial G}{\partial y}$ at all points in $U$.  
(This is the same as 
saying the vector field $(f,g) = \nabla G$, the gradient 
vector field for the function $G$, hence the name.)  
\begin{itemize}
\item[A)] (10) Show that the value of $G$ increases along
solution curves of the gradient system, and deduce that 
a gradient system never has nonconstant periodic solutions.
\item[B)] (10) What can you say about 
equilibrium points of gradient systems?
\end{itemize}
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\centerline{\it  Have a peaceful and joyous holiday season!}
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