Math 53 : Fall 2010

Course description

Differential equations are key in most applications of mathematics. They arise in such diverse fields as engineering, physics, biomathematics, chemistry, finance, and, obviously, mathematics itself. This course will introduce the basic definitions and methods used in the study of differential equations. We will focus especially on linear differential equations (and systems) and the methods specific to them. We will also see how understanding a linear system can allow us to get information about a (much more complicated) non-linear system.

From the course catalog :
Linear ordinary differential equations, applications to oscillations, matrix methods including determinants, eigenvalues and eigenvectors, matrix exponentials, systems of linear differential equations with constant coefficients, stability of non-linear systems and phase plane analysis, numerical methods, Laplace transforms. Integrated with topics from linear algebra (103).

Prerequisite: Math 51 (or equivalent). You should be familiar with differentiation and integration of functions, vectors, matrices, systems of linear equations, determinants, and inverses of matrices.

Course goals

By the end of the summer, you will:

• know what an ordinary differential equation is, what information it contains and what it means for a function to be a solution to a differential equation;
• understand what kinds of problems lend themselves to being studied by means of an ordinary differential equation;
• know several methods to extract information about a differential equation:
  1. by giving an explicit solution (when we are lucky and the equation is relatively simple)
  2. by using qualitative methods to find general properties of solutions
  3. by approximating solutions (using a computer) and knowing to be careful about limitations of the various techniques.
• know more advanced linear algebra techniques and theory, and how this applies to differential equations.

Meeting times

We meet every day (Monday, Tuesday, Wednesday, Thursday and Friday) from 2:15 PM to 3:05 PM, in room 200-303.

The summer quarter begins June 21st and the last lecture is August 12th. There is no class on July 5th.

The final exam is Friday, August 13th, from 12:15 PM to 3:15 PM.

Textbook

The textbook is Differential Equations : an Introduction to Modern Methods and Applications by James R. Brannan and William E. Boyce.

I will follow this textbook in the lectures (up to some chapter permutation), and will use it for some of the homework problems and extra credit problems.

A copy is on reserve at the mathematics library (4th floor of the math building).

New copies of the textbook come with a license to use ODE Architect, a piece of software for Windows that allows you to graph and compute with differential equations. See the front cover of the textbook and also the textbook's official site. This site also has study guides and other potentially useful resources. None of these are required. You may purchase a used textbook.

Course Grade

The course grade will be computed as follows :

• Midterm : 30 %
• Quiz average : 20 % (I will drop your lowest quiz score from the average)
• Project: 10 %
• Final exam : 40 %.

Homework will be assigned, but it will not be collected or graded. Instead, a weekly quiz will test you on one or two problems taken directly from the assigned homework. You may miss one quiz (if you take all the quizzes, I will drop the lowest quiz grade).

The midterm will be held in class on the 21st of July.

Projects: Part of your course grade will be determined by a group project. I will discuss this further on the first day of class.