Introduction

Calculus and Differential Equations first arose from some Physics problems that Sir Isaac Newton posed. Thus, the roots of differential equations are in applications to Mathematical models. Mathematical modeling is a broad area of Applied Mathematics that is interpreted in many different ways. My expertise is in Mathematical Biology and Dynamical Systems, particularly delay differential equations. This will clearly color my perspective of the subject. However, there are many general principles that follow in the modeling of any system, independent of the area of study or the mathematical technique employed.

So what is a mathematical model?

A mathematical model is a representation of a real system. The essence of a good mathematical model is that it is simple in design and exhibits the basic properties of the real system that we are attempting to understand. The model should be testable against empirical data. (The text uses an argument about modeling the population of rabbits in Wyoming.) The comparisons of the model to the real system should ideally lead to improved mathematical models. The model may suggest improved experiments to highlight a particular aspect of the problem, which in turn may improve the collection of data. Thus, modeling itself is an evolutionary process, which continues toward learning more about certain processes rather than finding an absolute reality. This use of mathematics is quite different from K-12 training in mathematics, where mathematics is treated as an absolute with exact answers.

Overview

This is a first course in differential equations, so we will be developing a number of techniques for solving differential equations. I have chosen the text by Blanchard, Devaney, and Hall because it uses what is known as the modern geometric approach to solving differential equations. Classical first courses in differential equations show a large number of specific techniques for solving differential equations. This course will present a more qualitative way at looking at differential equations with a heavier emphasis on the applications that generate the differential equations. I will also be showing you a number of computer techniques that will help in understanding the behavior of many of these differential equations.

The modern approach to differential equations fits into the larger context of general dynamical systems. Dynamical systems began with the advent of Calculus, which was formulated by Leibniz and Newton in the late1600s. Until recently, the emphasis has been on finding exact solutions to the differential equations which govern the dynamics of some modeling problem. However, early on, scientists discovered that analytical methods failed with problems as simple as the three body problem of gravitational attraction from astronomy. Before computers, scientists worked on clever methods to simplify the modeling problem to make it more tractable, usually using some approximations. The computational power of computers and the new software that is available means that a new approach is needed to learn differential equations.

In the late 1800s, Poincaré developed geometric methods to analyze mathematical problems. His perspective introduced the concept of qualitative analysis rather than quantitative analysis. Qualitative analysis looks into the behavior of dynamical systems and uses geometric techniques to predict what are the possible outcomes for certain nonlinear problems. Bifurcation theory developed to shown how behavior changed as certain parameters in the system changed. One of the outcomes of this analysis was the theory of chaos, which has been discussed widely in recent years since the popular book by James Gleick in 1987. The explosive growth in studying dynamical systems has been assisted by the high speed computer, which allows easy simulation of complex nonlinear problems. You will see that understanding the role of the parameters in the underlying dynamical systems is very important to understanding the possible outcomes of a model, and thus, whether it is an appropriate model.

A mathematical model that consists of differential equations provides only one direction of the modeling diagram given above. It is very important to be able to validate that mathematical model, which gets to the inverse problems where we must fit actual data and find the parameters in the model. This is often much more difficult and requires a different collection of mathematical tools. This course will concentrate on finding the forward solutions to the differential equations, but you should be aware that this is a small part of understanding the problem.

One study presented in the text is the damped spring (which we will analyze). Below is a graphical representation of this dynamical system.

More information on the damped oscillator and its solution in Maple with a movie of the solution curves rotating can be found by clicking on the picture. This movie is part of the advertisement for my Math 241 course on Maple.