Articles
This folder contains expository articles that advance the art and practice of the teaching and learning of ODEs.
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Integrating Factors and Repeated Roots of the Characteristic Equation
by Howard Dwyer, William Greenpublished January 19, 2012 -
Using Technology in a Differential Equations Course: Lessons Learned Implementing a New Paradigm
by Thomas Wanglerpublished October 14, 2011 -
A Project Approach in Differential Equations Courses
by Robert Borrelli, Courtney Colemanpublished July 26, 2011 -
Using Abel's Theorem to Explain Repeated Roots of the Characteristic Equation
by William Greenpublished July 26, 2011 -
The Laplace Transform: Motivating the Definition
by Howard Dwyerpublished February 7, 2011 -
Learning About ODEs Using Interactive Online Modules
by Kuva Jacobs, David Pantonpublished November 3, 2010 -
Teaching Introductory Differential Equations with ScicosLab
by Stephen Campbell, Ramine Nikoukhahpublished August 21, 2010 -
Classroom Voting: Active Learning in Differential Equations
by Kelly Cline, David Lomenpublished January 11, 2010 -
Pitfalls and Pluses in Using Numerical Software to Solve Differential Equations
by Robert Borrelli, Courtney Colemanpublished February 8, 2009
Most texts on elementary differential equations solve homogeneous constant coefficient linear equations by introducing the characteristic equation; once the roots of the characteristic equation are known the solutions to the differential equation follow immediately, unless there is a repeated root. In this paper we show how an integrating factor can be used to find all of the solutions in the case of a repeated root without depending on an assumption about the form that these solutions will...
Historically, a first course in Ordinary Differential Equations (ODEs) has been taught as a “methods course.” Namely, different types of differential equations are trotted out and the method of solution for each class of ODEs is presented. The student is then left to “master the method” by way of lengthy algebraic manipulations, a good dose of differential calculus, and of course, getting all the pieces of the new method done in the right order. All in all, it can be a daunting task...
From the late 60's the mathematics department at Harvey Mudd College (HMC) has been active in introducing independent study projects into its math courses, especially courses involving differential equations. This paper describes two such approaches and the features that were constructed to support them. With the change in technology in the late 90's, it was clear that these project approaches needed to be updated. These changes are underway and are described in this article.
This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel's Theorem.
Most undergraduate texts in ordinary differential equations (ODE) contain a chapter covering the Laplace transform which begins with the definition of the transform, followed by a sequence of theorems which establish the properties of the transform, followed by a number of examples. Many students accept the transform as a Gift From The Gods, but the better students will wonder how anyone could possibly have discovered/developed it. This article outlines a presentation, which offers a plausible...
This paper discusses the development and implementation of a set of online teaching and learning modules for the modeling and solution of simple linear Ordinary Differential Equations (ODEs). The paper describes how constructivist principles are used in the development of these modules and how in-built assessment has been used to enhance student learning. A discussion of the module structure is provided with particular emphasis on the concept of the module walk-through. Modules developed for the...
Perhaps the largest and most capable open source software for doing applied mathematics is the open source package Scicoslab. Widely used in Europe and Asia it is less well known in the United States. In this article we will explain how ScicosLab can be easily used to help teach differential equations and to also introduce students to a software package that they can take with them for the rest of their careers.
Classroom voting is a technique that involves students more deeply in the lesson being presented--the more the students are involved, the more they will understand, and the more they will retain. The key idea behind classroom voting is a series of multiple-choice questions within a lesson plan which is provided by the instructor. After each question is posed, the students are given a few minutes to work through the problem, to form an opinion, and then discuss the question in small groups before...
Ordinary differential equations (ODEs) are often used to model the behavior of physical phenomena and textbooks today especially demonstrate this fact. Since only a very small collection of ODEs can be solved analytically, there is often no alternative than to use computer software to gain some insight into the behavior of solutions (and sometimes even if solution formulas are available--the formulas are often complicated!). A classic work on the numerical solution of ODEs was authored by ...
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