Teaching a Modern ODE Course

Attending MAA MathFest 2014 in Portland, Oregon? Stop by this informational session on teaching a modern ODE course.

Hilton and Executive Tower

22nd Floor

Mount St. Helens Suite

Friday, August 8

3:30pm

The differential equations course has changed radically over the last quarter century.  Easy access to powerful computation has enabled visualization to play a much larger role. The increasing mathematization of the life sciences has greatly expanded the kinds of models available for investigation. The advent of dynamical systems has made new kinds of questions imaginable and accessible. A modern ODE course has to take all this progress into account, though it is perhaps not clear exactly how to do so.  Anne Noonburg―University of Hartford and author of OrdinaryDifferential Equations from Calculus to Dynamical Systems―and Steve Kennedy―Carleton College and MAA Books Sr. Acquisitions Editor―will lead a discussion focused on how best to react to these changes in your ODE course. We will ask such questions as:

• What is the appropriate role of modeling in the ODE course?
• How do we balance the needs of physics, biology and engineering majors in the course?
• What are good sources of deep and interesting models?
• How much emphasis on numerical methods is appropriate?
• Similarly how much dynamical systems theory and visualization?
• What is the appropriate role of technology and what are good choices?
• Is a Linear Algebra prerequisite necessary, or can the needed material fit in the ODE course?

New in the MAA Store: Ordinary Differential Equations

Ordinary Differential Equations: From Calculus to Dynamical Systems
By Virginia W. Noonburg
List Price: $60.00
MAA Member: $48.00

This book presents a modern treatment of material traditionally covered in the sophomore-level course in ordinary differential equations. While this course is usually required for engineering students the material is attractive to students in any field of applied science, including those in the biological sciences.

The standard analytic methods for solving first and second-order differential equations are covered in the first three chapters. Numerical and graphical methods are considered, side-by-side with the analytic methods, and are then used throughout the text. An early emphasis on the graphical treatment of autonomous first-order equations leads easily into a discussion of bifurcation of solutions with respect to parameters.

The fourth chapter begins the study of linear systems of first-order equations and includes a section containing all of the material on matrix algebra needed in the remainder of the text. Building on the linear analysis, the fifth chapter brings the student to a level where two-dimensional nonlinear systems can be analyzed graphically via the phase plane. The study of bifurcations is extended to systems of equations, using several compelling examples, many of which are drawn from population biology. In this chapter the student is gently introduced to some of the more important results in the theory of dynamical systems. A student project, involving a problem recently appearing in the mathematical literature on dynamical systems, is included at the end of Chapter 5.

A full treatment of the Laplace transform is given in Chapter 6, with several of the examples taken from the biological sciences. An appendix contains completely worked-out solutions to all of the odd-numbered exercises.

The book is aimed at students with a good calculus background that want to learn more about how calculus is used to solve real problems in today's world. It can be used as a text for the introductory differential equations course, and is readable enough to be used even if the class is being "flipped." The book is also accessible as a self-study text for anyone who has completed two terms of calculus, including highly motivated high school students. Graduate students preparing to take courses in dynamical systems theory will also find this text useful.

Order your copy today in the MAA Store.

MAA Books Beat: Extraordinary Book on Ordinary Differential Equations

Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUS. Extraordinary Book on Ordinary Differential Equations appears in the April/May 2014 issue.

Mathematics changes slowly; the mathematics curriculum changes even more slowly. A few years ago, to celebrate the tercentenary of L'Hôpital's Calculus, some colleagues and I read it seminar-style. The most striking thing about the experience was that his table of contents looked shockingly similar to our departmental calculus syllabus. That being said, in the 30 years I've been teaching collegiate mathematics, there is one course in the undergraduate math curriculum that has changed dramatically–the course in ordinary differential equations (ODE).

These changes are rooted in the calculus reform movement of the late 1980s and early 1990s, the easy access to powerful computing and visualization, and the rise of dynamical systems theory and its accompanying qualitative point of view.

The calc reform movement taught us to explain everything from graphical, numerical, and symbolic perspectives. Computing, of course, made it possible to do this in effective ways, especially the numerical and graphical bits. Dynamical systems provided entirely new ways of thinking about the evolution and bifurcation of systems.

The New on View

All these changes are fully on view in Virginia (“Anne”) Noonburg's Ordinary Differential Equations from Calculus to Dynamical Systems, newly released by the MAA. Noonburg has a distinguished record of research in dynamical systems, especially concentrating on equations that model biological systems. You clearly see these intellectual interests in this book.

New Textbook Coming Soon

Ordinary Differential Equations: From Calculus to Dynamical Systems

by V. W. Noonburg

Catalog Code: FCDS

This book presents a modern treatment of material traditionally covered in the sophomore-level course in ordinary differential equations. While this course is usually required for engineering students, the material is attractive to students in any field of applied science, including those in the biological sciences.

The standard analytic methods for solving first and second-order differential equations are covered in the first three chapters. Numerical and graphical methods are considered side-by-side with the analytic methods and are then used throughout the text. An early emphasis on the graphical treatment of autonomous first-order equations leads easily onto a discussion of bifurcation of solutions with respect to parameters.

The fourth chapter begins the study of linear systems of first-order equations and includes a section containing all of the material on matrix algebra needed in the remainder of the text. Building on the linear analysis, the fifth chapter brings the student to a level where two-dimensional nonlinear systems can be analyzed graphically via the phase plane. The study of bifurcations is extended to systems of equations using several compelling examples, many of which are drawn from population biology. In this chapter, the student is gently introduced to some of the more important results in the theory of dynamical systems. A student project involving a problem recently appearing in the mathematical literature on dynamical systems is included at the end of chapter 5.

A full treatment of the Laplace transform is given in Chapter 6, with several of the examples taken from the biological sciences. An appendix contains completely worked-out solutions to all of the odd-numbered exercises.

The book is aimed at students with a good calculus background who want to learn more about how calculus is used to solve real problems in today's world. It can be used as a text for the introductory differential equations course, and is readable enough to be used even if the class is being “flipped.” The book is also accessible as a self-study text for anyone who has completed two terms of calculus, including highly motivated high school students. Graduate students preparing to take courses in dynamical systems theory will also find this text useful.

To see all of our textbooks visit the MAA website.