Here are some ideas for possible projects. Four further suggestions:
a) look through the books on reserve at the library  b) Come into my office
and take a look at some of the projects from former years c) ask a
professor in one of your other courses for ideas  d)  come prepared to 
tell me something about your interests and ask me. * topics are those which
have lead to undergraduate theses.


When I do not make a suggestion on where to start in the literature,
you can find the subject in any differential equations course. Probably
the best source of information is a web search, concentrating on web
pages for undergraduate and beginning graduate courses.


1.  Electrical circuits   For the electrical engineers in the course, if
you would like to write a project about how ODEs are used in your sublect,
this is fine.  The more ambitious might like to discuss Van der Pol's
equation as well.


2.  The Laplace transform.  We will not do this in this course. If you
are going to need it in your engineering courses, this would be a good
idea.

3*.  Ideas about chaos.  Devaney's book is a good place to start. Topics
for projects could be:  Period doubling, Cantor sets, symbolic dynamics,
complex functions and the definition of the Julia set.  Professor Williams
would be a good reference.

4*.  The Lorenz attractor This is a 3X3 system of equations which has chaotic
behavior.  References include the book by Strogatz on reserve,  our text,
and the references in these books.  Professor Williams is also a good
reference for this. Solutions have knotted periodic orbits.

5. The double pendulum.  Many of you have seen the double pendulum in my
office.  The challenge is to work out the equations.  This need a good
knowledge of physics.

6. Planetary motion using the central force model.  When asking physicists
about a good reference for this, they refer me to Feynman's lectures. I
think it is volume 1.* Write an ODE solver for satelites. Unfortunately,
the member of the astronomy department who used to help is retired to Maine.


7. The Hopf bifurcation  This is discussed in Strogatz with a lot of examples.
I believe it is also in Hirsch-Smale (although my copy has vanished and I
don't know for sure).

8.  The Poincare Bendixson theorem.  Hirsch and Smale, Strogatz, all classsical
books   on ODE. This is a theorem about periodic orbits in non-linear planar
systems.

9.  The curve of matrices  exp (At) were A is an nxn matrix.  Hirsch and Smale
and many books on equations intended for mathematicians. If you are studying
matrices, you can also investigate the relationship between A and exp(A).

10*. The matrix groups SU(2) and S0(3). The first is the special unitary group
of complex 2x2 matrices. The second is the matrix group of  3x3 rotations.
I am not sure I have an elementary reference.  You could try your physics
book.

11*. The Hodgkin-Huxley and Fitzhugh-Nagumo Model for nerve impulses. It
isn't often that math gets a Nobel Prize.  This is in Murray, Edelstein-Keshet
and most of the biological references.

12*.  Epidemiology...the study of spread of disease. Edelstein-Keshet, Murray

13*.  Population dynamics, preditor-prey and competition,  almost every
book on ODE.  The Volterra-Lotke equations are quite famous.  We will
talk about them in class.

14*.  Enzymes.  The Michaelis-Menten equations.  Murray,  Segel, Edelstein-
Keshet.

15. Models for detection and treatment of Diabetes, Differential Equation
Models by Braun.  This is a fairly standard topic in texts.

16.  Traffic Flow theory,  This uses partial differential equations, but
there are several reasonable projects here.  (Braun, see above).

17.  Hilbert's (unsolved) 16th problem.  This is an unsolved problem
posed by the famous mathematician Hilbert concerning the number of
closed orbits of equations which come from polynomials in the plane.
(Braun, see above).

18.  Special Functions (of mathematical physics, but mathematicians use
them as well).  Bessel functions, elliptic functions, Hermite, Laguarre
and Legendre polynomials all can be defined using differential equations.
These functions have many important and beautiful properties.  A classic
text like Coddington and Levinson as well as many books on mathematical
methods in physics treat these subjects.  At one time, we used to teach
them in the equivalent of 427K, but no more. Your book, for all its
faults, partially treats them at the end of Chapter 5. Elliptic functions
are very important in algebraic geometry, which is high level math at its
best.

19*. Mathematics and music.  I will give one lecture on this at the end of
the course, and I have a collection of books and articles in my office.

20. Game theory.  This should be a topic of a course, but if you just want 
to get an idea of what a Nash equilibrium is, it makes a good topic. Another
Nobel prize winning topic in math.

21. Computer modeling. The dull one is to write an ODE solver.Once there was
a theory that anybody who was anybody did this at least once. I never did.

22*. Computer modeling. Write a code for systems of difference equations and
develop some graphics to code information about the system.