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Recent documents in CODEE Journalen-usFri, 18 Jan 2019 00:53:37 PST3600Experiences Using Inquiry-Oriented Instruction in Differential Equations
https://scholarship.claremont.edu/codee/vol11/iss1/3
https://scholarship.claremont.edu/codee/vol11/iss1/3Tue, 13 Nov 2018 11:33:46 PST
Student-centered instruction can be a challenging endeavor for teachers and students. This article reports on the use of the Inquiry-Oriented Differential Equations (IO-DE) curriculum (Rasmussen, 2002) in an undergraduate differential equations course. Examples of student work are shared with specific reference to research in mathematics education.
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Keith NabbTeaching Differential Equations without Computer Graphics Solutions is a Crime
https://scholarship.claremont.edu/codee/vol11/iss1/2
https://scholarship.claremont.edu/codee/vol11/iss1/2Tue, 13 Nov 2018 11:33:37 PST
In the early 1980s computer graphics revolutionized the teaching of ordinary differential equations (ODEs). Yet the movement to teach and learn the qualitative methods that interactive graphics affords seems to have lost momentum. There still exist college courses, even at big universities, being taught without the immense power that computer graphics has brought to differential equations. The vast majority of ODEs that arise in mathematical models are nonlinear, and linearization only approximates solutions sufficiently near an equilibrium. Introductory courses need to include nonlinear DEs. Graphs of phase plane trajectories and time series solutions allow one to see and analyze the crucial behaviors, whether or not analytic solutions exist. Furthermore, interactivity is key to experimenting with parameters in order to modify behaviors. Now, a quarter of a century later, we have far more technology--but many features of the original software have been lost in the rush to the future. We have both educational and software concerns. This is not only an academic issue--scientists at multiple nonacademic agencies (e.g., FDA, NIH, USCGS) immediately took up our software tools in the late 1980s, and increasingly more of our students come from fields that did not traditionally require mathematics background. We should not be depriving today's students of the skills to analyze behaviors of solutions to ODEs.
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Beverly H. WestTeaching an Online Sophomore-Level Differential Equations Course with Mathematica Supplements
https://scholarship.claremont.edu/codee/vol11/iss1/1
https://scholarship.claremont.edu/codee/vol11/iss1/1Mon, 11 Jul 2016 19:45:28 PDT
I have had the experience of developing and teaching a number of sections of online sophomore-level differential equations courses for the past eight years. This article is an attempt to recall my methods, the ideas and philosophies that guided me, give an informal summary of student achievement and course evaluations, and describe my creation and use of interactive Mathematica supplements in the course. Four of the supplements that I created with Mathematica are available in the online appendix to this article.
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William M. KinneyTeaching Differential Equations with Graphics and without Linear Algebra
https://scholarship.claremont.edu/codee/vol10/iss1/3
https://scholarship.claremont.edu/codee/vol10/iss1/3Mon, 11 Jul 2016 19:25:33 PDT
We present our approach to teaching the Method of Eigenvectors to solve linear systems of ODEs without assuming a prerequisite course in Linear Algebra. Rather we depend heavily on a graphical approach to systems in two dimensions to motivate the eigenvalue equation.
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Nishu Lal et al.A Field Guide to Programming: A Tutorial for Learning Programming and Population Models
https://scholarship.claremont.edu/codee/vol10/iss1/2
https://scholarship.claremont.edu/codee/vol10/iss1/2Mon, 11 Jul 2016 19:25:30 PDT
Programming skills and concepts are best taught within an applied framework in the students' discipline. However, many tutorials teach the skills and concepts, but alienate the applications and usefulness. We have produced a Field Guide to Programming, a tutorial that uses the discrete time population growth model as a concrete example to introduce and explain programming concepts. We equate our Field Guide to the beginning chapters of any naturalist's field guide, where the use of the guide is explained. This Field Guide covers a range of topics from simple mathematical expressions and assigning variables to functions and solvers for ordinary differential equations. We wrote and have used this Field Guide individually for self learners, as introductory and supplementary material for courses, as the outline for workshops, and a guide for multiple hands-on recitations within a course. After working through this Field Guide either alone or in a workshop setting, students will have the conceptual background to begin to use programming as a problem-solving tool and the terminology to begin to read programming-specific tutorials. We have included two versions of the tutorial: one for use with MATLAB and tested for compatibility with Octave and one for use with the R programming language. We have also included script files of the code from this Field Guide.
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Christopher Stieha et al.Modeling the Effects of Avian Flu (H5N1) Vaccination Strategies on Poultry
https://scholarship.claremont.edu/codee/vol10/iss1/1
https://scholarship.claremont.edu/codee/vol10/iss1/1Mon, 11 Jul 2016 19:25:27 PDT
The work in this article addresses a problem posed by Dr. Maria Salvato to the CODEE community. The task was to model costs associated with varying vaccination strategies for the Avian Flu virus (H5N1) on chicken populations. The vaccination strategies proposed included vaccination varying proportions of the flock with live virus vaccine, dead virus vaccine, and no vaccination. This article encompasses the construction of a model for the problem using a modification to the SIER model and the subsequent analysis of that model. The analysis of the model revealed the most cost effective vaccination strategy to be vaccination of half the flock with dead virus vaccine.
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Cooper J. Galvin et al.Integrating Factors and Repeated Roots of the Characteristic Equation
https://scholarship.claremont.edu/codee/vol9/iss1/13
https://scholarship.claremont.edu/codee/vol9/iss1/13Mon, 11 Jul 2016 17:10:30 PDT
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 take. We also show how an integrating factor can be used to explain the "extra" power of t which appears in the trial form of the solution when using the method of undetermined coefficients on a nonhomogeneous equation in the case where the right hand side is a polynomial multiple of the corresponding homogeneous solution.
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Howard Dwyer et al.A Simple Charged Three-Body Problem
https://scholarship.claremont.edu/codee/vol9/iss1/12
https://scholarship.claremont.edu/codee/vol9/iss1/12Mon, 11 Jul 2016 17:10:27 PDT
The dynamics of a simple model of three charged bodies interacting under an inverse square electrostatic force is presented. The model may be viewed as an alternative to the pendulum, the standard model of a periodically forced and damped nonlinear oscillator.
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Jaie Woodard et al.Linear Operators and the General Solution of Elementary Linear Ordinary Differential Equations
https://scholarship.claremont.edu/codee/vol9/iss1/11
https://scholarship.claremont.edu/codee/vol9/iss1/11Mon, 11 Jul 2016 17:10:25 PDT
We make use of linear operators to derive the formulae for the general solution of elementary linear scalar ordinary differential equations of order n. The key lies in the factorization of the linear operators in terms of first-order operators. These first-order operators are then integrated by applying their corresponding integral operators. This leads to the solution formulae for both homogeneous- and nonhomogeneous linear differential equations in a natural way without the need for any ansatz (or "educated guess"). For second-order linear equations with nonconstant coefficients, the condition of the factorization is given in terms of Riccati equations.
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Norbert EulerDelay-Differential Equations with Constant Lags
https://scholarship.claremont.edu/codee/vol9/iss1/10
https://scholarship.claremont.edu/codee/vol9/iss1/10Mon, 11 Jul 2016 17:10:22 PDT
This article concerns delay-differential equations (DDEs) with constant lags. DDEs increasingly are being used to model various phenomena in mathematics and the physical sciences. For such equations the value of the derivative at any time depends on the solution at a previous "lagged" time. Although solving DDEs is similar in some respects to solving ordinary differential equations (ODEs), it differs in some rather significant ways. These differences are discussed briefly. The effect the differences can have on systems of ODEs and DDEs is illustrated. Popular approaches used in the development of numerical methods for solving DDEs are described. Available Matlab DDE solvers and a Fortran 90 solver based on these approaches are mentioned. Finally, some pointers to further resources available to interested readers are given.
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Lawrence Shampine et al.Evolution of the Modern ODE Course
https://scholarship.claremont.edu/codee/vol9/iss1/9
https://scholarship.claremont.edu/codee/vol9/iss1/9Mon, 11 Jul 2016 17:10:19 PDT
The rapid development of technology in the latter part of the twentieth century has revolutionized the teaching of differential equations. In this paper we will try to trace the evolution of this important change. We tried to include the most important efforts in this regard, but we apologize in advance if some efforts have slipped our attention.
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Beverly West et al.Understanding Differential Equations Using Mathematica and Interactive Demonstrations
https://scholarship.claremont.edu/codee/vol9/iss1/8
https://scholarship.claremont.edu/codee/vol9/iss1/8Mon, 11 Jul 2016 17:10:16 PDT
The solution of differential equations using the software package Mathematica is discussed in this paper. We focus on two functions, DSolve and NDSolve, and give various examples of how one can obtain symbolic or numerical results using these functions. An overview of the Wolfram Demonstrations Project (http://demonstrations.wolfram.com) is given, along with various novel user-contributed examples in the field of differential equations. The use of these Demonstrations in a classroom setting is elaborated upon to emphasize their significance for education.
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Paritosh Mokhasi et al.Agent-Based Fabric Modeling Using Differential Equations
https://scholarship.claremont.edu/codee/vol9/iss1/7
https://scholarship.claremont.edu/codee/vol9/iss1/7Mon, 11 Jul 2016 17:10:13 PDT
We use an agent-based modeling software, NetLogo (http://ccl.northwestern.edu/netlogo), to simulate fabric drape by applying a modified mass spring system. This model provides an application of harmonic motion to textiles and fashion, fields not typically discussed in the undergraduate differential equations classroom. Euler's method is coded into the model to solve a system of ordinary differential equations describing the fabric's position over time. Our interactive NetLogo model allows students to visualize the behavior of the system and to experiment with parameters. We show an example of the success of our program. Students can also experiment with other numerical methods.
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Joseph Rusinko et al.Using the Taylor Center to Teach ODEs
https://scholarship.claremont.edu/codee/vol9/iss1/6
https://scholarship.claremont.edu/codee/vol9/iss1/6Mon, 11 Jul 2016 17:10:10 PDT
This article introduces a powerful ODE solver called the Taylor Center for PCs (http://www.ski.org/gofen/) as a tool for teaching and performing numeric experiments with ODEs. The Taylor Center is an All-in-One GUI-style application for integrating ODEs by applying the modern Taylor Method (Automatic Differentiation). The Taylor Center also offers dynamic graphics (including 3D stereo vision). After a brief review of the features of the Taylor Center, we consider instructive examples of ODEs in various applications and also several particular examples illustrating intricacies of numeric integration. The article therefore continues the thesis of Borrelli and Coleman (CODEE Journal, http://www.codee.org/ref/CJ09-0157) that awareness and caution are needed while interpreting the results of numeric integration. We offer practical ideas and advice on how to use the Taylor Center for teaching ODEs, and to increase the motivation and interest of students.
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Alexander GofenThe Laplace Transform: Motivating the Definition
https://scholarship.claremont.edu/codee/vol8/iss1/5
https://scholarship.claremont.edu/codee/vol8/iss1/5Mon, 11 Jul 2016 16:20:24 PDT
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 (hopefully) progression of thoughts, which leads to integral transforms in general, and the Laplace transform in particular.
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Howard DwyerODE Architect: Now and Future
https://scholarship.claremont.edu/codee/vol8/iss1/4
https://scholarship.claremont.edu/codee/vol8/iss1/4Mon, 11 Jul 2016 16:20:21 PDT
This article describes current compatibility issues with ODE Architect software suite and 64-bit Windows operating systems, and proposes viable alternatives using virtual machine environment and cloud computing.
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Jeho ParkUsing Abel's Theorem to Explain Repeated Roots of the Characteristic Equation
https://scholarship.claremont.edu/codee/vol8/iss1/3
https://scholarship.claremont.edu/codee/vol8/iss1/3Mon, 11 Jul 2016 16:20:18 PDT
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.
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William GreenA Project Approach in Differential Equations Courses
https://scholarship.claremont.edu/codee/vol8/iss1/2
https://scholarship.claremont.edu/codee/vol8/iss1/2Mon, 11 Jul 2016 16:20:16 PDT
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.
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Robert Borrelli et al.Using Technology in a Differential Equations Course: Lessons Learned Implementing a New Paradigm
https://scholarship.claremont.edu/codee/vol8/iss1/1
https://scholarship.claremont.edu/codee/vol8/iss1/1Mon, 11 Jul 2016 16:20:12 PDT
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 for students and it usually does little to pique the interest or curiosity of most students. This is the old paradigm. In an attempt to make the course more interesting and less daunting, a new paradigm was adopted – a paradigm that incorporates technology as a core component of the course, rather than just an “add on.” This article examines various issues related to implementing this new paradigm as well as some lessons learned along the way.
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Thomas WanglerPitfalls and Pluses in Using Numerical Software to Solve Differential Equations
https://scholarship.claremont.edu/codee/vol7/iss1/5
https://scholarship.claremont.edu/codee/vol7/iss1/5Mon, 11 Jul 2016 14:55:19 PDT
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 (Numerical Solution of Ordinary Differential Equations, Chapman and Hall/CRC,1994). There are some questions about the behavior of solutions of ODEs that are not quite appropriate for numerical solvers. In this paper we present examples which illustrate some of these features. However, there is no disputing the fact that the output of numerical solvers is often useful for portraying and understanding the behavior of solutions of ODEs and their utility in modeling physical phenomena, as our final example shows.
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Robert Borrelli et al.