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PrefacePreface

Coordinated Differential Equations is adapted from Thomas W. Judson's Open Educational Resource (OER) The Ordinary Differential Equations Project, originally published under a GNU Free Documentation License in 2018. This text was modified from the original to better fit the curriculum of MATH 221Differential Equations at the University of Nebraska - Lincoln.

While the spirit of the original text is preserved in this version, there are some structural differences. Perhaps most significantly, the treatment of second-order linear equations now immediately follows the introductory material in Chapter 1. In Judson's original, the second chapter is devoted to linear systems of equations; second-order equations are covered later in the text using techniques from linear systems. (In particular, the homogeneous second-order equation \(ax'' + bx' + cx = 0\) is solved by introducing an extra variable \(u = x'\text{,}\) which transforms the original equation into a two-by-two linear system.) In Coordinated Differential Equations, second-order linear equations are solved solely using the characteristic polynomial, with all linear algebra material deferred to later chapters.

In order to give a richer picture of solutions to second-order linear equations, a new section in Chapter 1 is devoted entirely to complex numbers and their role in differential equations. This section includes basic information on complex arithmetic, as well as deeper material on Euler's formula (which is derived by solving the complex initial value problem \(z' = iz\text{,}\) \(z(0) = 1\)) and various trigonometric identities. The chapter on the Laplace Transform also occupies a more prominent place in Coordinated Differential Equations than in the original, immediately following Chapter 2 on second-order linear equations.

While some exercises have been slighlty modified, and a few new ones added, the vast majority of the exercises in Coordinated Differential Equations are identical to those in The Ordinary Differential Equations Project. One caveat in this regard is with respect to exercises for which a computer algebra system may be useful or necessary: the original text contains many "built-in" SageMath modules, while the modified text does not. This is due to the fact that Sage is not commonly used in UNL mathematics courses, MATLAB being the more typical choice. As such, most references to Sage were removed during the developent of Coordinated Differential Equations. (Of course, interested students are still encouraged to explore other tools, including Sage, on their own!).

Judson's original preface is included below for completeness. The present authors are heavily indebted to his work in developing the original text, which gives a clear, concise, and highly engaging introduction to ordinary differential equations.

Austin Eide and Adam Larios
The University of Nebraska - Lincoln
Lincoln, Nebraska 68588

This text is intended for an undergraduate course in ordinary differential equations. The Ordinary Differential Equations Project began when the author was teaching the ordinary differential equations course at Harvard University. After arriving at Stephen F. Austin State University, the Harvard notes began to transform into the makings of a textbook. At the same time, the author was converting his abstract algebra book, Abstract Algebra: Theory and Applications (http://abstract.pugetsound.edu/index.html) from LaTeX into MathBook XML. With MathBook XML, which is now PreTeXt (https://pretextbook.org), one can produce HTML, PDF, EPUB, and even braille versions of a textbook while only having to maintain the PreTeXt source.

There has been a strong trend during the past few decades to incorporate both modeling and technology into undergraduate differential equations courses. Since it is easy to insert computational cells inside an HTML version of the textbook with PreTeXt, there is now an opportunity to seemlessly embed technology into the textbook. Sage (sagemath.org), our technolgy of choice, is a free, open source, software system for advanced mathematics. Sage is ideal for assisting with a study of ordinary differential equations, since it cannot only be embedded as computational cells in a textbook, it can also be used on a computer, a local server, or on CoCalc (https://cocalc.com). The Sage code in The Ordinary Differential Equations Project has been tested for accuracy with the most recent version available at this time: Sage Version 9.2 (released 20201024).

There are additional exercises or projects at the end of many of the sections. Many of the projects come from SIMIODE (https://www.simiode.org). SIMIODE provides a rich environment for learning and teaching differential equations through modeling. SIMIODE was founded by Dr. Brian Winkel, Emeritus Professor of Mathematics, United States Military Academy, West Point NY USA in 2013. Some of the projects may require a basic knowledge of programming. All of these exercises and projects are more substantial in nature and allow the exploration of new results and theory.

Another great source of problems and projects is the CODEE Journal, a peer-reviewed, open-access publication, distributed by the CODEE (Community of Ordinary Differential Equations Educators) and published by the Claremont Colleges Library (https://scholarship.claremont.edu/codee/). The goal of the CODEE Journal is to advance the teaching and learning of ODEs through the dissemination of materials that will be useful to both educators and education researchers.

Thomas W. Judson
Stephen F. Austin State University
Nacogdoches, Texas 75962

History

  1. The Differential Equations Project, Thomas W. Judson, 2018.
  2. Coordinated Differential Equtaions, Austin Eide and Adam Larios, 2023.