MATHEMATICS 2250-3

Ordinary Differential Equations and Linear Algebra

Fall semester 1998

Prerequisites:

Course Outline:

We will cover most of chapters 1-8 in the Edwards-Penney text, as well as chapter 5 from the Gustafson-Wilcox text. If you intend eventually to take ``Partial Differential Equations for Engineers'', you should buy the entire Gustafson-Wilcox text since it is the book used in that course. Even if you do not plan on taking Math 3150 later, the Gustafson-Wilcox book provides another viewpoint to the material in Edwards-Penney and you may wish to purchase it as an additional text for this course. You are only required to purchase the chapter 5 supplement, however.

The course begins with first order differential equations, a subject which you touched on in Calculus. Recall that a differential equation is an equation involving an unknown function and its derivatives, that such equations often arise in science, and that the order of a differential equation is defined to be the highest order derivative occurring in the equation. The goal is to understand the solution functions to differential equations since these functions will be describing the scientific phenomena which led to the differential equation in the first place. In chapters one and two of Edwards-Penney we learn analytic techniques for solving certain first order DE's, the geometric meaning of graphs of solutions, and the numerical techniques for approximating solutions which are motivated by this geometric interpretation. We will carefully study the logistic population growth model from mathematical biology and various velocity-acceleration models from physics.

In chapter 3 we will study the theory of higher-order linear DE's, and focus on the second order ones which describe basic mechanical and electrical vibrations. You were introduced to these in Calculus as well, but we will treat them more completely now. We will study forced oscillations and resonance in this setting. Then in chapter 4 we will introduce systems of (several) differential equations (for several related unknown functions) and indicate how these arise naturally in dynamical systems in which it takes several functions to describe the complete behavior. We will study numerical methods for approximating solutions but will delay the analytic theory (chapters 5-6) until the latter part of the course.

Our next topic will be the Laplace transform and its applications to the study of linear DE's, chapter 7. This ``magic'' transform takes differential equations and ``transforms'' them into algebraic equations. You will have to see it in action to appreciate it. You will see, for example, that this method gives a powerful way to study forced oscillations in the physically important cases that the forcing terms are step functions or impulse functions. Finally, in chapter 8, we will introduce our last technique for studying (single) differential equations, the method of power series. It is based on the idea of Taylor series, which you saw in Calculus.

At this point in the course we will take a month-long digression to learn the fundamentals of linear algebra - a field of mathematics which we need to understand in order to talk meaningfully about the theory of higher order linear DE's and of systems of linear DE's. We will use the chapter 5 supplement from Gustafson-Wilcox. The chapter starts out with matrix equations and the Gauss-Jordan method of solution. When you see such equations in high-school algebra you might be thinking of intersecting lines in the plane, or intersecting planes in space, or ways to balance chemical reactions, but the need to understand generally how to solve such equations is pervasive in science. From this concrete beginning we study abstract vector spaces and linear operators. Not only is such abstract theory useful in studying linear maps (or differentiable maps) between Euclidean spaces, but it is the framework which allows us to understand solution spaces to systems of linear differential equations as well. Along the way we will discuss the subtopics of determinants, eigenvalues and eigenvectors.

After the linear algebra digression we return to chapter 5 of Edwards-Penney and apply our theory to understand the solutions of systems of linear differential equations. We then study fundamental modes and resonance questions for forced oscillations in complicated mechanical systems like multi-story buildings. Throughout the course we will be showing how linear differential equations approximate (more accurate) nonlinear ones near points of equilibrium. In chapter 6 we will study global phenomena for nonlinear DE's, including phase-plane analysis and the notion of chaos.

There will be four computer projects assigned during the semester, related to the classroom material. They will be written in the software package MAPLE. There is a Math Department Computer Lab at which you all automatically have accounts, and there are other labs around campus where Maple is also available, for example at the College of Engineering. There will tutoring center support for these projects (and for your other homework) as well. The Math Department Lab and Engineering Math Tutoring Center are both located in Building 129, between JWB and LCB.

You will be permitted to work your solutions to the computer projects in different languages, such as Mathematica or Matlab, if you prefer these, although it will be easier to find help if you are using Maple. Notice that the text and the Computing Projects supplement support all three of these language however.

Grading:

It is the Math Department policy, and mine as well, to grant any withdrawal request until the University deadline of October 23.

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