The Ordinary Differential Equations Project

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.html²) from LaTeX into MathBook XML. With MathBook XML, which is now PreTeXt (https://pretextbook.org³), one can produce HTML, PDF, EPUB, and even braille versions of a textbook while only having to maintain the PreTeXt source. The Ordinary Differential Equations Project is now available on https://runestone.academy⁴, a Learning Engineering and Analytics Portal (LEAP) for hosting textbooks, assignments, and interactive learning activities.

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.org⁵), 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.com⁶). 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 2020–10–24).

There are additional projects at the end of each chapter. Many of the projects come from SIMIODE (https://www.simiode.org⁷). 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/⁸). 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.

Each section in The Ordinary Differential Equations Project contains reading questions. Reading questions should be answered by students before they come to class. The questions serve two purposes: students will have a general knowledge of the section to be covered in class before the actual class and instructors will have a better idea of what their students actually know. If instructors use Runestone, students can answer the reading questions in the textbook, where instructors can read and grade the student responses.

The Ordinary Differential Equations Project contains classroom activities in each section. The classroom activities can used in a variety of ways. I divide my class into groups of three or four and have each group work on the activities on the board. This allows me to efficiently comment on student work. Groups can also easily present their work to the rest of the class.