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{\bf Math 325, Spring 2001}
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  {\bf Tips for Studying for the Final}
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 The final is Wednesday, May 9, 4:15-6:15.  It includes an a take-home
 part, which will be posted before class on Monday, April 30, and due at
 the start of the in-class part.  The final will
cover everything we have done, so Sections 4.1-4.4, 8.1, 8.3, 8.5,
 Chapter 7,  Sections 9.1-9.3, Sections 10.1-10.7, and Chapter 6
 in Boyce and DiPrima and
Chapters 5, 7, 8, and 11-13
of {\it Differential Equations with Maple}.
You may bring a summary (both sides of an  $8\frac 12'' \times 11''\/$
sheet of paper, with notes in your writing) to the final.


Since you are allowed to bring in notes,
I am not testing your ability
to memorize formulas, algorithms, rules, etc. 
As you know by now, it is easy to get a computer to solve many ODE and
systems of ODE, often explicitly, and almost always numerically.  That
means that the ability to do computations by hand is much less
important. An understanding of the theory, an ability to
tell whether computer output is reasonable, and the ability  to interpret the
output
is very important.  

I also assume that you
learned techniques of integration in second semester calculus, got
good at doing them 
 when you did multiple integrals in third semester
calculus (if you weren't already good at them), and refreshed your
ability to do them on homework problems in this course, so I don't
need to test your ability to carry out a skill you learned long ago.  
I will provide you with a copy of the table of Laplace transforms from
page 300 of Boyce and DiPrima and also a table of Fourier transforms
of some standard functions. 

What should you expect the in-class part of the exam to look like?
It will have 6 problems, some with several parts, each worth 15-20
points. 
(There will also be a 10 point bonus problem.)  Almost 
\begin{math}
 \frac 23
\end{math}
of the points will be on material not included on the
midterm---nonlinear systems, Fourier series, the heat equation, the
wave equation, and the Laplace transform.  As I mentioned several
times in class, there wouldn't be much point to doing the Laplace
transform just to give you another way of solving the kinds of
problems you solved in Chapters 3 and 4.  Because of my mathematical
interests, I have a particular fondness for the heat equation. 

Some of the problems will involve computing solutions.
What kind of problems have you learned how to compute solutions for
this semester?   Constant coefficient
 linear equations (higher order and using  the Laplace transform), first order 
constant coefficient systems, the heat
 equation and the wave equation.  I won't ask you to compute a numerical
solution---that is best done by computer.   


Since the take-home exam tests your ability to write Maple code,
I won't ask you to write any during the in-class exam.
 

How can I test your ability  to
tell whether computer output is reasonable and if so, to interpret it?
I will guarantee at least one problem (and at most two)
 in which you are
shown a Maple worksheet which has the Maple input and output but no
comments.  The worksheet will have some relationship to problems you
have done.  You will  be asked various questions connected with
the worksheet.  Since you can't find explicit solutions for nonlinear
systems, but you can get a lot of information about solutions from 
plots of various kinds, nonlinear systems are good candidates for 
Maple problems.  How can you study for such problems? Although the
Maple problems on the midterm were on linear systems and numerical
solutions, start by making sure you understand those.  Then make sure
you understand things we've done with computers and nonlinear systems,
such as some of the problems in Problem Set F and Project 1.

While some of the problems on the exam will require an understanding of theory,
or be a lot easier if you do understand some theory, I won't test
that understanding very directly.  By that, I mean
there will be no problems of the type that often cause you the most 
trouble in homework---none involving reading abstract mathematical
statements or doing proofs.  However, you can count on at least one
problem which will require you to use ideas from the course in a new way.

 
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