Here are some ideas for possible projects. Five 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 talk to me, e) look more carefully
at parts of the text which are not on the syllabus for more ideas. Polking's
book on reserve has projects. 


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.Most web sources are
meant either for the general public (not mathematical enough) or too
technical (too mathematical?). Be careful about what you learn from
Wikopedia.  It is really useful but not completely reliable.

1) Here is a list of suggested topics from a textbook by Polking, Bogess
, Differential Equations with Boundary Value Problems 2nd edition.

Project 3.5 (p 132)  The spruce budworm.
Project 4.8 Nonlinear Oscillators (p 187).
Project 8.6 (p370) Long-Term behavior of solutions
Project 9.11 Oscillation of Linear Molecules
Project 10.9 Human Immune Response to Infectious Disease


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.  The Lorenz attractor This is a 3X3 system of equations which has chaotic
behavior.  References include the book by Strogatz on reserve as well as our
text. 

4. Basic enzyme kinetics.  This is agreat project for the biochemists in
the course. The best reference is Edelstein-Keshet. Murray and Keener and
Sneyd are advanced references.

5. The double pendulum.Come see the double pendulum in my office. 
The challenge is to work out the equations.  This need a good
knowledge of physics. In fact, I can't do it without something called
Lagrangian mechanics.  Some physics students have been able to do it
with advanced courses under their belt.

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. A harder alternative is the two body problem. The
goal is to start with Newton's differential equations and obtain the
elliptical orbits, as well as the equal area for equal time law.


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. Models for detection and treatment of Diabetes, Differential Equation
Models by Braun.  This is a fairly standard topic in texts. I think I can find
more references if you ask me.

9.  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).

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. Keener and Sneyd also do it.

12.  Epidemiology...the study of spread of disease. Edelstein-Keshet, Murray
Britton. We will talk about the SIRS model in class, but you can go further
and discuss  the subject in more detail.  There is something to say abou
epidemics like the h1n1 we were so afraid of last year.


13.  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. Boyce and DiPrima 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.

14. 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.

15.  Population models.  We will discuss very briefly the competitive model
and the preditor-prey model in class.  One project is to do this carefully
and work out some examples.