# Materials to Support the Teaching and Learning of Ordinary Differential Equations

## Printed and Print-Related Materials

### Inquiry-Oriented Differential Equations (IODE)

IODE is a first course in differential equations focused on understanding of the big ideas in first order, second order, nonlinear, and systems of differential equations, taught using an inquiry-oriented approach. The course is designed as a full semester course and topics covered include solving ODEs; numerical, analytic and graphical solution methods; solutions and spaces of solutions; linear systems; linearization; qualitative analysis of both ODEs and linear systems of ODEs; structures of solution spaces.

### CODEE Newsletters

From 1992–1997, CODEE ("Consortium of ODE Experiments" was our former name), with generous support from the National Science Foundation, published a newsletter that provided a regular source of ideas, inspiration, and experiments for instructors of ODEs. These newsletters have been archived by the Claremont Colleges Digital Library.

### Differential Equations Laboratory Workbook

Originally published in 1992, this 368-page workbook includes a plethora of classroom ready exercises and projects for students in differential equations courses. Using an approach that closely parallels what goes on in science and engineering laboratories, this workbook provides computer experiments that amplify topics found in introductory ordinary differential equations texts. Excellent 2 and 3-D graphics illustrate the range of qualitative behavior of solutions and give compelling visual evidence of theoretical deductions and a greater understanding of the qualitative properties. The experiments are largely self-contained and are independent of any particular hardware/software platform or text.

This book is made available for free with permission from John Wiley and Sons.

### Spotlights on ODE Topics

In preparation for the second edition of the their ODE textbook “Differential Equations: A Modeling Perspective,” Bob Borrelli and Courtney Coleman were asked to focus every section by removing material that was optional. This material was then gathered up into special sections called Spotlights and some of these sections were put at the end of chapters in the text. The remaining 27 Spotlight sections were put on a CD-ROM (along with ODE Architect) which was distributed along with the text.

These Spotlights describe different applications of ODEs and elaborate on techniques and concepts that play a role in the mathematical modeling process. Every Spotlight has a problem set at the end and are suitable for self-study, student project work, or even class lectures.

These 27 Spotlights have been made available for free with permission from John Wiley and Sons.

### A Dynamic Dynamical Systems Classroom

Glen Van Brummelen has made available 23 in-class activities for a differential equations course on topics ranging from population models to glucose and insulin in the bloodstream. All of the activities are available as PDF or Microsoft Word files.

### JMM 2013 Minicourse Materials

Michael Huber, Tom LoFaro, Ami Radunskaya, Daniel Flath, and Darryl Yong organized a minicourse at the 2013 Joint Mathematics Meetings in San Diego entitled "Teaching Differential Equations with Modeling." The materials from that minicourse are archived at the link above. Included in the materials is a book with 18 classroom-ready projects involving ODEs.

### Bibliography of Papers on the Teaching and Learning of ODEs

In 2011, Karen Marrongelle prepared a bibliography of 61 research articles on the teaching and learning of differential equations. These papers were compiled by searching the ERIC (US Government interface) database, Academic Search Complete, Dissertations and Masters theses, Education Full Text, Informaworld, PsychINFO, and JSTOR. The RCME volumes and Conference on Research in Undergraduate Mathematics Education proceedings were searched by hand for papers on the learning and teaching of differential equations. Of the 61 articles, 36 were published in peer-reviewed journals; the rest were from dissertations or conference proceedings.

### ODE Reviews

Tom Judson has compiled a list of short reviews of articles relating to ordinary differential equations. Many of them relate to the teaching and learning of ODEs, or describe interesting applications of ODEs that could be useful in a course on ODEs.

## Software

### Slopes

Developed by a group of faculty and students at Pepperdine University, Slopes is an interactive iOS app for exploring graphical solutions to ordinary differential equations. It consists of five activities with preloaded examples: Slopefields, Phase Planes, Systems, Waves, and Methods.

### Slope and Direction Fields

Slope and Direction Fields is a Javascript-based web app that allows for the visualization of slope fields and solutions to single-variable and planar ODEs. It is maintained by Darryl Nester at Bluffton University.

### ODEToolkit

ODEToolkit is a Java program that helps users calculate, visualize, and explore solutions to ordinary differential equations. It is made freely available for non-commercial download and use by Harvey Mudd College.

### dfield and pplane

These two Java programs allow students to interact and analyze ODEs without having to learn a complicated syntax or programming language. They are made available for free for use in educational institutions.

### DESSolver

DESSolver is a Java-based ODE system solver that allows students to compute and display solutions.

### OdeFactory

OdeFactory allows students to visualize dynamical systems with dimension four or less. The systems can be studied as ordinary differential equations and as discrete iteration maps. OdeFactory can generate appropriate phase space images, or "views," of the dynamical systems.

### Mathematical Visualization Toolkit

Although the Mathematical Visualization Toolkit is designed to help students better visualize the concepts of Calculus, it also includes several tools related to differential equations. It has the ability to numerically calculate and graph the solution to arbitrary differential equations with one, two, or three state variables.

### Interactive Differential Equations

The Interactive Differential Equations (IDE) website is specifically designed for students taking a differential equations course. It is remarkably easy to use and focuses on helping students visualize the mathematics. This site contains more than 90 interactive differential equations tools and covers the entire differential equations course: first-order differential equations, second order differential equations, linear and nonlinear applications, Laplace Transforms, series solutions, and boundary value problems. Applications are drawn from engineering, physics, chemistry, and biology.

### The Taylor Center

The Taylor Method ODE Solver is distinguished from other numerical methods in that only the Taylor Method computes the increments of the solution with theoretically unlimited precision so that the integration step need not approach zero even for an arbitrary degree of accuracy. This is possible because the method uses automatic differentiation--exact computing of the derivatives up to any desired order--to obtain the Taylor polynomial of any degree of the solution components.

### MIT Mathlets

The MIT Mathlets constitute a suite of carefully designed and highly interactive Javascript applets designed to enhance selected university level STEM classes. Some of them touch on topics relating to dynamical systems and differential equations.