Exam III will be in the usual classroom on Friday, April 20, 2011. It will cover sections 4.1, 5.1-5.2, and 7.1-7.2, but only the material that we discussed in class, and obviously not every single item can be covered in depth in a 50-minute exam. The current draft has 51 points possible.

Be sure you follow the instructions of each problem, and give the answers requested, but do not waste time doing things that are not required. As on any exam, it is wise to start with the problems that you feel confident that you know how to do, before moving on to others. Most of the questions will have rather short solutions, if you know how to do them, so if you find yourself doing something lengthy and unusually complicated on a problem, it might be best to move on to other problems and come back later to it later if you have time.

Calculators are not needed, although you may, if you really want to, use a non-graphics simple arithmetic calculator without even trig functions or log and exponential, but no mechanical or electronic device more sophisticated that this (this includes iPods, earpieces, etc.). Blank paper will be provided, so all you will need is something to write with. As on the quizzes, please write your solutions on the blank paper. You may have as many sheets as you need, and may put the problems in any order. Please put your name on your exam paper and all pages of your solutions, and hand them all in together, although without any pages of scratch work that is not to be graded.

Some of the exam problems will be very similar to homework problems, while others will draw upon the material presented in the lectures. Definitions of important concepts are perfectly reasonable questions, and although you do not need to know them word-for-word, you should be able to write down a coherent and accurate definition of any major concept or term.

The emphasis is on first-order linear systems, the eigenvalue method, and Laplace transforms, although the theory in section 4.1 underlies the methodology of sections 5.1-5.2. The following topics are very likely to appear, although the exam is not limited to them:

1. Basic notation and theory of matrices, determinants and their properties, calculating determinants by expansion along rows or columns.
2. Solving linear systems by Gauss-Jordan elimination.
3. First-order linear systems of differential equations, their solutions and general solutions, the matrix viewpoint.
4. Eigenvalues and eigenvectors of matrices, what they are, how to calculate them, and how they give us solutions to certain kinds of first-order linear systems.
5. The IVP for first-order linear systems, solving IVP's using Gauss-Jordan elimination.
6. The definition of the Laplace transform of a function, using it to compute Laplace transforms.
7. Finding Laplace transforms and inverse Laplace transforms using tables of Laplace transforms and Laplace transform properties.

You do not need to know Cramer's Rule or, for this exam, the method of partial fractions. It is not necessary to memorize Laplace transforms of specific functions or general properties of the Laplace transform, as one page of the exam will be this list of Laplace transform formulas. Of course, the more familiar you are with these formulas, the better you will be able to apply them.

As usual, exams from some of my previous differential equations classes can be found at my course pages page. At that time, however, the course syllabus did not include matrix and eigenvalue methods (but did cover power series methods), so for this particular exam the previous exams are not very relevant.