Calculus I,II. That is, a thorough knowledge and understanding of differential and integral calculus in one variable. Some acquaintance with the basic notions of physical science, for the purposes of motivating and applying differential equations. Calculus III (multi-variable) and Linear Algebra are helpful, but not required.
Differential equations (DEs) form the backbone of mathematical modeling throughout modern science. They are important and interesting because they are the expression of physical, biological, economic or other laws, whenever those laws concern the rates of change of interrelated variables. For instance, DEs describe mechanical oscillations, chemically reactions, electrical circuits, ecological population dynamics, and much more. Their solutions are the quantitative predictions that follow from the scientific laws.
In this course we study only ordinary differential equations, which are DEs whose unknowns depend on one variable. Most often the independent variable represents time. But the dependent variable may represent all sorts of different quantities. When there is just one dependent variable we refer to a differential equation, and when there are several dependent variables (or equivalently a dependent vector) we refer to a system of differential equations.
Our goal in learning about differential equations and systems is two-fold. First, we must understand how these mathematical models arise from problems in science and engineering. Then, we must find ways to solve the equations, in order to extract the predictions of the models. This interaction between modeling and analysis is a key feature of the subject, and we will constantly go back and forth between formulating equations and calculating their solutions.
The textbook takes a modern approach, which emphasizes qualitative thinking about solutions as well as calculating them analytically. To facilitate this the textbook puts all differential equations in the form of first-order systems. For instance, an oscillator governed by a second-order equation becomes a coupled system of two first-order equations. We will learn how to visualize solutions of systems, especially systems of two equations which can be portrayed on a plane. In addition, we will develop analytical solution methods for systems, especially linear systems; in the process we will learn the necessary methods from linear algebra.
We will consider numerical methods very briefly in this course, even though they are important whenever analytical techniques are not available. They are discussed in detail in another course, Math 551. For the purposes of illustration in class, I will show computer-generated solutions and portraits. But students will not be required to implement numerical methods.
It is essential to do a variety of problems when learning mathematics.
Two kinds of homework problems will be assigned regularly as we cover topics. We will use the WebWork system for most of these problems, particularly the more routine questions about solution technique. Early in the semester I will explain the logistics of using this web-based homework system. But we also need to work on modeling-type problems that require complete written solutions. For those problems, which will be few in number but more substantial in content, the homework will be handed in and read by a human grader.
Always keep up to date with your reading and re-reading of the textbook. It should be clear which section we are going to discuss each day of class, and you are strongly encouraged to read that section before the class. That way you will get the most out of our presentation and discussion. Then shortly after class you should re-read the section in the textbook, connect its presentation to the lecture, and tackle the assigned homework problems. In advance of any examination, you should consolidate your knowledge and try a few extra problems to test it.