The final exam will be cumulative, although the majority of the questions will be drawn from material covered since the most recent midterm exam. There will be several questions based only on material from before the second midterm.

The following is a list of the major topics which we have covered since the second midterm exam.

Topics:

Here are more detailed descriptions of some of these ideas and concepts.

(1) You should know how to solve systems of constant-coefficient differential equations by the direct method of elimination (analogous to the standard method for solving systems of linear algebraic equations) as well as by the eigenvalue method. For the former method, you should be able to use the language of differential operators fluently (see section 4.2).

Make sure you are familiar with the basic concepts and terms of linear algebra (see pages 284-290 for a review) and that you can convert between a system of differential equations and a single equation expressed in matrix form. Also, be able to find eigenvalues and eigenvectors for a given matrix and to use this knowledge to construct a full set of linearly independent solutions to a given system of equations.

Here are a few sample problems for you to try. In each case, find the general solution to the system of equations x'=Ax, where A is the given matrix.

1. 1	2
   -1	4

2. 3	2	2
   -5	-4	-2
   5	5	3

3. 1	-4
   4	9

4. 1	0	2
   2	3	-4
   1	1	0

5. 1	0	0
   1	4	3
   2	-3	-2

6. 0	0	1
   0	1	-4
   0	1	-3

7. 2	0	0	0
   -21	-5	-27	-9
   0	0	5	0
   0	0	-21	-2

(2) Know what the Laplace transform of a function is and how to calculate it. You should be familiar with the rules for computing the Laplace transform of the derivative (Theorem 1, page 457).

Be able to solve a differential equation or system of equations by using the Laplace transform. I will supply you with a brief table of some of the most common Laplace transforms, but you may have to do some algebraic work to be able to use the entries in the table. Know how to use partial fractions to decompose a rational function into a sum of rational functions with low-degree denominators. Be familiar with the basic properties of the Gamma function (equations (7), (8) and (9) on page 447); you may find it necessary to use these facts to simplify your solution to a problem.