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Modules/Chapters 10-12 apply two-dimensional systems to models that illustrate more advanced techniques and theory; the multimedia approach makes them nevertheless quite accessible. The modules are intended to enable students to get much further with the technical aspects explained in the chapters than would be otherwise possible.
Module/Chapter 13 treats discrete dynamical systems in an introductory fashion that could be used in a course in ODEs, calculus, or even a non-calculus course.
A Multimedia appendix on numerical methods gives insight into the ways in which numerical solutions are constructed.
Description/Prerequisites for Individual Modules/Chapters
We list below for each Module/Chapter its prerequisites and some comments on its level and goals. In general, each module progresses from easier to harder submodules, but the first section of nearly every module is at an introductory level.
The modules can be accessed in different orders. It is not expected that they will be assigned in numerical order. Consequently, we have tried to explain each concept wherever it appears, or to indicate where an explanation is provided. For example, Newton’s second law, F = ma, is described every time it is invoked.
There is far more material in ODE Architect than could possibly fit into a single course.
Module/Chapter 1: Modeling with the ODE Architect
Assumed concepts: Precalculus; derivative as a rate of change
This module is unlike all the others in that it is not divided into submodules, and it provides a tutorial for learning how to navigate ODE Architect. It carries that tutorial process along in tandem with an introduction to modeling that assumes very little background.
Module/Chapter 2: Introduction to ODEs
Assumed concepts: Derivatives; slopes; slope fields
The module begins with some simple first-order ODEs and their solutions and continues with slope fields (and a slope field game).
The Juggler and the Sky Diver submodules use second-order differential equations, but both the chapter and the module explain the transformation to systems of two first-order differential equations.
Module/Chapter 3: Some Cool ODEs
Assumed concepts: Basic concepts of first-order ODEs, solutions, and solution curves
Newton’s law of cooling, and solving the resulting ODEs by separation of variables or as linear equations with integrating factors, are presented thoroughly enough that there need be no prerequisites.
The submodule for Cooling a House extends Newton’s law of cooling to real world cases that are easily handled by ODE Architect (and not so easily by traditional methods). This section makes the point that rate equations and numerical solutions are often a much smarter way to go than to trudge toward a solution formula.
Module/Chapter 4: Second-Order Linear Equations
Assumed concepts: Euler’s formula for complex exponentials
The module and chapter treat only constant coefficient ODEs. The chapter begins by demonstrating how to treat a second-order ODE as a system of first-order ODEs which can be entered in ODE Architect. Both the first submodule and the chapter explain from scratch all the traditional details of an oscillating system such as amplitude, period, frequency, damping, forcing, and beats.
The Seismograph submodule is a real world application. The derivation of the equation of motion is not simple, but the multimedia module gives insight into the workings of a seismograph, and it is not necessary to understand the details of the derivation to use and explore the modeling ODE.
Module/Chapter 5: Models of Motion
Assumed concepts: Newton’s second law of motion
This module’s collection of models of motion in one and two dimensions is supported by a chapter that gives background on vectors, forces, Newton’s laws, and the details of the specific submodules; so it stands on its own without further prerequisites.
Module/Chapter 6: First-Order Linear Systems Assumed concepts: Basic matrix notation and operations (multiplication, determinants); complex numbers; Euler’s formula
This unit introduces all of the basic notions, both algebraic (emphasized in the chapter) and geometric (emphasized in the module), for linear systems. The central roles of eigenvalues and eigenvectors are explained. The Tool can be used to calculate eigenvalues and eigenvectors.
The Explorations bring in coupled tank problems (Chapter 8 introduces compartment models) and small motions of a double pendulum (which are extended in Chapter 7).
Module/Chapter 7: Nonlinear Systems
Assumed concepts: Equilibrium points; phase plane and component plots; matrices; eigenvalues and eigenvectors
The goal is to use graphical solutions to make handling nonlinear systems as easy (almost) as linear systems. Linearization of a nonlinear ODE is introduced as a basic concept, and the chapter goes on to elaborate perturbations and bifurcations. The Tool can be used to find equilibrium points, and calculate the Jacobian matrix and its eigenvalues/eigenvectors at each equilibrium point. The predator-prey and saxophone reed models are introduced and explained in the module while the spinning bodies and double pendulum models are treated in the chapter and also in the Library with an animated model linked to the ODE.