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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.
Module/Chapter 8: Compartment Models
Assumed concepts: Systems of ODEs
Both the module and the chapter use 1D, 2D, 3D, and 4D applications (in sequence) to illustrate principles of the Balance Law and interpretations of solutions. The final submodule introduces Hopf bifurcations and the interesting behavior of chemical reactions in an autocatalator. Three of the models are linear; the last is nonlinear.
Module/Chapter 9: Population Models
Assumed concepts: Systems of ODEs
The module and chapter introduce simple 1D, 2D, and 3D nonlinear models, and give a discussion of the biology behind the models.
Module/Chapter 10: The Pendulum and Its Friends
Assumed concepts: Systems of ODEs; the first submodule of Module 4; the arctangent function; parametric curves on a surface
The pendulum submodule explores all the traditional aspects of a pendulum, using integrals of motion. Child on a Swing and Geodesics on a Torus give new extensions of pendulum analysis; supporting detail is given in the chapter. The approach to modeling is a little different in this chapter—for example, how to invent functions that behave as needed (Child on a Swing), or how to exploit part of an ODE that looks familiar (Geodesics on a Torus).
Module/Chapter 11: Applications of Series Solutions
Assumed concepts: Systems of ODEs; acquaintance with infinite series and convergence; the first submodule of Module 4
The module introduces the techniques and limitations of series solutions of second-order linear ODEs. The Robot and Egg provides motivation for the subject and Aging Springs illustrates Bessel functions. The chapter contains information about the mathematics of series solutions.