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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Review Sheet -- Exam 1}
\centerline{September 28, 2000}
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\noindent
{\it General Information}
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The first midterm exam this semester will be given in class on 
Thursday, October 5, as announced in the course syllabus.  This 
will be a closed-book exam, but you may use calculators at
any point, and I will provide a copy of the table of integrals
from the endpapers of our textbook for your use during the 
exam.  

The exam will cover the material from Chapters 1 - 3 of the text
that we have studied since the start of the semester (see specific
topics and review problems below).  

If there is interest, I would be happy to arrange a late afternoon
or evening review session for the exam.  Tuesday, October 3 is probably
the best day to aim for.
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\noindent
{\it Topics To Be Included}
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\item{1)}  ODE, solutions, initial value problems, the Existence
and Uniqueness Theorem for solutions of first order initial value problems
\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.
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\noindent
{\it Suggested Review Problems}
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\item{} From \S 1.2: 16, 17, 18, 20, 21, 23 - 28
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\item{} From \S 1.3: 1, 3, 4, 5
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\item{} From \S 1.4: 3, 4, 5 (Note: a ``sink'' is an equilibrium with
the property that all solutions that start with $y$ near the equilibrium
level tend {\it toward} that level as the independent variable 
tends to $+\infty$; a ``source'' is an equilibrium with
the property that all solutions that start with $y$ near the equilibrium
level tend {\it away from} that level as the independent variable 
tends to $+\infty$.  Equilibria that are neither sinks nor sources
are called ``nodes''.)
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\item{} From Review Problems at end of Chapter 2 (page 87):  try 
a good selection of these.  Of course the first issue is to decide which
class of equation you have and what method(s) apply.
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\item{} From \S 3.2:  1, 3, 7, 9 (the ``Malthusian law'' is ${dP\over dt} = kP$
for some constant $k$), 13, 18
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\item{} From \S 3.3: 1, 3, 7, 11 (Note: in the ODE
$${dT\over dt} = K(A(t) - T) + H(t) + K_U(T_D - T)$$
representing ``proportional control'' heating and air conditioning 
($T_D = $ desired temperature, $H(t)$ is the heat added from sources
other than the heating/ac system, $A(t)$ is the
ambient temperature), the ``time constant'' with heating/ac
is ${1\over K + K_U}$; the time constant for the building or van 
without heating/ac is ${1\over K}$.
\bye