Differential equations is an old but durable subject that remains alive and useful to a wide variety of engineers, scientists, and mathematicians. The purpose of this book is to provide an introductory graduate text for these consumers. It is intended for classroom use or self-study. The goal is to provide an accessible and concrete introduction to the main principles of ordinary differential equations and to present the material in a modern and rigorous way. The intent of this goal is to provide the solid foundation that will enable a reader to learn and understand other parts of the subject easily and to encourage them to learn more about differential equations and dynamical systems.
The study of differential equations began with the birth of calculus, which dates to the 1660s. Part of Newton’s motivation in developing calculus was to solve problems that could be attacked with differential equations. For example, an early triumph of differential equations was Newton’s demonstration that Kepler’s empirical laws of planetary motion could be derived from Newton’s laws of motion using differential equations. Now, with over 300 years of history, the subject of differential equations represents a huge body of knowledge including many subfields and a vast array of applications in many disciplines. It is beyond exposition as a whole. Instead, the right question to ask is what are the principles of differential equations that a serious user should know and understand today?
Principles of Differential Equations is my answer to this question. It looks at ordinary differential equations from the viewpoint of important principles. Although the word “principle” is probably overused in the academic world and may be a bit trite, it is used here seriously in the sense of “a basic or essential quality or element determining intrinsic nature or characteristic behavior.” Each section presents a coherent picture of a circle of ideas that illustrates a key principle in the study of differential equations. The overarching questions driving the theory are discussed and the value and limitations of results are explained. Throughout, the book a concerted effort is made to tie the pieces together and give the reader a coherent and unified sense of the subject.
Principles of Differential Equations is also largely about the qualitative theory of ordinary differential equations. Qualitative theory refers to the study of the behavior of solutions without determining explicit formulas for the solutions. It originated with Poincare at the beginning of the twentieth century and, in my judgment, has been the most important theme of ordinary differential equations in that century. Consequently, very little attention is paid to techniques for finding analytic formulas for solutions. The emphasis is on the general properties of the solutions of ordinary differential equations from simple existence of solutions to the remarkable behavior of Hopf bifurcations.
1. Fundamental Theorems
2. Classical Themes
3. Linear Differential Equations
4. Constant Coefficients
6. The Poincare Return Map
7. Smooth Vector Fields
8. Hyperbolic Phenomenon
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