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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Review Sheet -- Final Examination}
\centerline{November 30, 2000}
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\noindent
{\it General Information}
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\item{$\bullet$} The final examination for this class will be given during the 
scheduled period -- 2:30 to 5:30 pm on Thursday, December 14.  
\item{$\bullet$}  The final will be a {\it comprehensive exam}, covering
all the topics from the two midterms, {\it and the material about
linearization of 1st order autonomous systems, solutions of 1st order
linear systems with constant coefficients, and classification of
critical points from the last 3 weeks.}  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{$\bullet$}  Like the second in-class midterm, this will be an
{\it open book exam}.  This means you do not have to memorize 
everything, {\it but it also means that I can ask a ``curve ball''
question where you have to think a bit (take what you know and apply 
it in a new situation).}
\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 November
30.
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\noindent
{\it Topics To Be Included}
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\noindent
The exam will cover the following topics from Chapters 1-5, 9, 12 of 
the text:
\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)}  Qualitative theory for 1st order ODE: Direction Fields, 
special techniques for autonomous equations (identifying equilibrium
solutions, determining whether they are ``stable'' or ``unstable'' 
-- see section 1.4 in the text and Discussion 1)
\item{3)}  Techniques for deriving analytic solutions of 1st order 
ODE:  Separable equations, linear equations, exact equations, integrating
factors, substitutions and transformations for homogeneous equations,
equations of the form ${dy\over dx} = g(ax + by + c)$, Bernoulli equations.
\item{4)}  1st order ODE in modeling of mixing problems, heating/cooling, etc.
\item{5)}  The theory of second order linear ODE:
the existence and uniqueness theorem
for solutions, fundamental sets of solutions for homogeneous equations,
the Wronskian and Abel's identity, general solutions of inhomogeneous equations.
\item{6)}  The constant coefficient case -- the auxiliary equation,
forms for solutions when the roots are real or complex.
\item{7)}  Superposition and the method of undetermined coefficients
for inhomogeneous equations.
\item{8)}  Applications to mass-spring systems, electrical circuits, etc.
Resonance phenomena.
\item{9)}  First order systems -- critical points,
the phase plane, trajectories and integral curves.
\item{10)}  First order systems with constant coefficients -- solutions
via eigenvalues, eigenvectors of the coefficient matrix.
\item{11)}  Linearization of a first order system at a critical point,
classification of critical points in terms of eigenvalues of the
linearized system (see sections 12.2, 12.3 in text)
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\noindent
{\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:
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\item{} Section 9.5/17, 18, 19, 21, 25, 35, 37, 42, 43
\item{} Section 9.6/1, 3
\item{} Section 12.2/1, 7
\item{} Section 12.3/9, 11
\bye