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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Review Sheet -- Final Examination}
\centerline{December 6, 2004}
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{\it General Information}
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\item{$\bullet$} The final examination for this class will be given during the 
scheduled period -- 8:30 to 11:30 am on Thursday, December 16.  
\item{$\bullet$}  The final will be a {\it comprehensive exam}, covering
all the topics from the two midterms, {\it and the material about
Hamiltonian systems and the undamped and damped pendulum
systems from after the second midterm.}  See the list of topics
below for more details.
\item{$\bullet$} The exam will be written to take about 2 hours if you work 
steadily, but you will have the full 3 hour period to use if you need that 
much time.  
\item{$\bullet$}  The exam will be closed-book, calculators allows,
and I will provide a table of integrals.
\item{$\bullet$}  If there is interest, I would be happy to arrange
an evening review session during exam week (Monday or Tuesday would 
probably be the best days).  We can discuss this in class on December
6.
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{\it How to Prepare}
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\noindent
The best way to prepare for this exam will be to do (a lot of)
specific problems related to this material.  You will want to
have the facility
to do the calculations necessary without spending a lot of time
deciding how to get started.  We have done almost all of the 
relevant problems in the Hirsch, Smale, Devaney book, though ( :( )
So, to study, I strongly suggest looking at problems from
the Blanchard, Devaney, Hall book (on reserve in the Science Library).
Chapters 1, 3, 4, 5 cover the same material that we have discussed (plus a
few additional topics)
and every section has lots of exercises!
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{\it Topics To Be Included}
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\noindent
\item{1)}  Terminology of ODE, solutions, initial value problems, the Existence
and Uniqueness Theorem for solutions of first order initial value problems
(one equation, one dependent variable).
\item{2)}  Techniques for deriving analytic solutions of 1st order 
ODE:  Separable equations, linear equations
\item{3)}  Qualitative theory for 1st order ODE: Direction Fields, 
special techniques for autonomous equations (identifying equilibrium
solutions, determining whether they are ``sinks'' or ``sources'' or
neither)
\item{4)}  1st order ODE in modeling of mixing problems, heating/cooling, etc.
\item{5)}  Bifurcation diagrams for 1-parameter families of ODE.
\item{6)}  First order systems $X' = AX$ with constant coefficient
matrix -- solutions
via eigenvalues, eigenvectors of the coefficient matrix.  The 
canonical forms for higher-dimensional systems, and how
to find solutions based on the canonical form.
\item{7)}  The trace-determinant plane and the classification 
for $2\times 2$ systems.  Bifurcations in families.
\item{8)}  Applications to unforced and forced oscillator equations:
$$mx'' + bx' + kx = g(t),$$
mass-spring systems, electrical circuits, etc.
Resonance phenomena.
\item{9)}  Non-linear first order systems -- equilibrium points,
the phase plane, nullclines, solutions
\item{10)}  Linearization of a first order system at a critical point,
classification of critical points in terms of eigenvalues of the
linearized system.
\item{11)}  Hamiltonian systems, their special properties, applications
to the undamped and damped pendulum equations.
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{\it Suggested Review Problems}
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See review sheets for Midterm Exams 1 and 2 for topics 1-9 in the list
above.  For topics 10, 11, from Blanchard, Devaney, Hall:
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\item{} Section 5.1/3, 11, 17
\item{} Section 5.2/1, 3
\item{} Section 5.3/3 (Hint: solve for $x$ to sketch the level curves), 
9, 11, 13, 

\bye