This chapter is organized in much the same way as the first half of Chapter 1. It begins with a section on modeling in which we return to the predator-prey models mentioned in Section 1.1. We also discuss the undamped harmonic oscillator. The next three sections are devoted to introducing our three-pronged approach in terms of the study of systems of differential equations. Section 2.2 introduces geometric and qualitative concepts, Section 2.3 presents some analytic techniques that apply to special systems, and Section 2.4 describes Euler's method as it is applied to systems. Unlike Chapter 1, we do not feel the need for a separate section on existence and uniqueness theory, so the Existence and Uniqueness Theorem is presented at the end of Section 2.4. Section 2.5 begins the qualitative discussion of the Lorenz system as an introduction to three-dimensional systems. This last section is the first part of a discussion of three-dimensional systems that reappears in Sections 3.8, 5.5, and 8.5. These four sections can be safely omitted if necessary. (At Boston University, we often omit these sections due to time constraints.)

Although all of Sections 2.1-2.4 contain material that is consistently used throughout the remainder of the text, it is conceivable that one could skip to Chapter 3 immediately after Section 2.2.

The use of technology to draw vector fields, direction fields, and phase portraits is essential. In order to begin to get a feel for what pictures to expect, students must see many examples accurately drawn. HPGSystemSolver often fits the bill. The CD also contains other special purpose tools that are useful at various times (see the comments below).

2.1 Modeling via Systems

The transition to systems is natural for students. Relating the solution curve in the phase plane to the graphs of the components is considered difficult by some but is generally mastered after sufficient effort. (More practice to help develop this skill is provided in Section 2.2.) It is important for students to realize that both types of graphs are necessary because neither graph alone contains all of the information about a solution. A good analogy is trying to understand a slinky from its shadows (see Figure 2.16). Bring a slinky to class. It can be used both to demonstrate a mass-spring system and to illustrate Figure 2.16.

The section is based on predator-prey models and the undamped harmonic oscillator. We focus on these two models because we feel that these are two of the best to use at this point. In the case of the predator-prey equations, we delay an explanation of how we obtain the graphs of solutions until Section 2.4. For the undamped harmonic oscillator we simply guess solutions. At this point, we do not have enough theory to discuss the general solution. Many of the exercises focus on interpretation---both of the equations in a system and of the behavior of the solutions.

DETools

In addition to the general purpose HPGSystemSolver, the special purpose tools PredatorPrey and MassSpring provide easy-to-run classroom demonstrations that are very relevant to this section.

Comments on selected exercises

Exercises 1 and 15 involve the interpretation of the parameters in a system while Exercises 9-14 involve the interpretation of the equations.

Exercises 2-6 require an analysis of a predator-prey system similar to the one carried out in the section.

Exercises 7, 8, 16, and 17 give practice in going from the phase plane to the graphs of component functions and in the interpretation of solutions. Exercise 17 is particularly good for assigning an essay. (This predator-prey phenomenon really does occur.)

In Exercises 9-14 and 18, modifications are made to a predator-prey model. Modifying an existing model is easier than developing models from scratch, but it is still challenging. In Exercise 18, there is more than one reasonable answer.

Exercises 19 and 20 provide practice with solutions of equations for undamped harmonic oscillators. Both of these exercises prepare the students for ideas that are discussed in more detail in Sections 2.2 and 2.3.

Exercises 21-24 are modeling problems that involve Hooke's law applied to mass-spring systems.

In Exercises 25-30, models for concentrations of reactants in simple chemical reactions are developed. These models reappear in Section 5.2 (Exercises 16-20).

2.2 The Geometry of Systems

DETools

In addition to the general purpose HPGSystemSolver, the special purpose tools DESketchPad and GraphingSolutionsQuiz help students understand the relationship between a solution curve in the phase plane and the corresponding x(t)- and y(t)-graphs.

Comments on selected exercises

Exercises 1-6 involve vector fields and phase portraits for very simple systems.

Exercises 7 and 8 involve vector fields and phase portraits for simple second-order equations.

In Exercises 9 and 10, we provide the direction field, and the student sketches the phase portrait. A description of the long-term behavior of a particular solution is also requested.

Exercises 11 is a "matching" exercise reminiscent of the matching exercise in Section 1.3. This type of exercise is an excellent way to test understanding of geometric concepts such as slope fields and direction fields. If brief explanations are required to justify why a given system corresponds to a particular direction field, students must examine the fields closely.

In Exercise 12, students find equilibrium solutions for a predator-prey system discussed in Section 2.1.

In Exercises 13-18, students are expected to solve for the equilibrium points by hand, but then they produce the phase portraits with the aid of a computer. One important goal is to give them practice using paper-and-pencil techniques and technology in the same exercise. Exercise 19 is a similar exercise for a second-order equation (an example of Duffing's equation).

In Exercise 21, students match solution curves with their corresponding x(t)- and y(t)-graphs.

Exercise 22 is one of our favorite "animated" homework problems.

In Exercises 23-26, students must produce x(t)- and y(t)-graphs from solution curves in the xy-phase plane. Their answers should include a scale on the x,y-axis, but they cannot provide a scale on the t-axis.

Exercise 27 goes the opposite direction as Exercises 23-26. The x(t)- and y(t)-graphs are given, and the student must draw the corresponding solution curve. In theory, doing this exercise is fairly mechanical, but try it before you assign it.

Exercises 28 and 29 are meant to motivate the study of uniqueness. These two exercises are repeated (exactly) as Exercises 12 and 13 in Section 2.4. Logically they make more sense in Section 2.4, but they are also stated here in case you want to get your students thinking about these issues before uniqueness is discussed.

2.3 Analytic Methods for Special Systems

Using the method of undetermined coefficients from Section 1.8 or integrating factors from Section 1.9, we also show how one can derive the general solution of a "partially" decoupled linear system. Knowing how to solve this class of systems is useful when we study repeated eigenvalues in Chapter 3, and it provides the students with a systematic method for deriving the solutions to a system.

We also take this opportunity to introduce damped harmonic oscillators in this section. This model is used throughout the text, so the terms and equations presented on pages 193 and 194 are extremely important. Since this section is devoted to techniques that produce closed-form solutions, we also introduce the technique of guessing solutions of the form y(t) = e^st. The Linearity Principle does not appear until Section 3.1, so we stop short of finding general solutions of the form y(t) = k₁ e^{s₁ t} + k₂ e^{s₂ t}. However, we do make the connection between solutions of the form y(t) = e^st and the corresponding vector-valued solution Y(t) = (y(t), v(t)) = (e^st, s e^st).

DETools

Some of the solutions obtained in this section can be illustrated with MassSpring tool, and HPGSystemSolver can be used to plot phase portraits and x(t)- and y(t)-graphs.

Comments on selected exercises

Exercises 1-4 provide routine practice at checking functions to see if they are solutions.

Exercises 5-12 involve a given partially decoupled system. They illustrate the technique introduced in this section, and they provide additional practice with the geometric ideas introduced in Section 2.2.

Exercises 13-14 involve the analysis of a mass-spring system that involves two springs.

Exercises 15-18 provide the same type of practice for second-order equations. These equations reappear in Exercises 21-24 of Section 3.2.

Exercise 19 is a nonlinear partially decoupled system.

2.4 Euler's Method for Autonomous Systems

If Section 1.4 was supplemented with material from Chapter 7, then it is also possible to supplement this section with material from Section 7.3.

As has already been mentioned, it is also convenient to discuss existence and uniqueness theory for systems at this point.

The example of a swaying building is presented in this section because quantitative information determines which model is more appropriate.

DETools

EulersMethodForSystems can be used in class to demonstrate the method, and the students are encouraged to use this tool when they answer Exercises 3-6.

Comments on selected exercises

Exercises 1 and 2 involve computing Euler's method solutions with fairly large step sizes and comparing the results with the direction field and/or actual solutions. The computations are tedious but manageable if done by hand, but the expectation is that the students will use some type of technology such as a spreadsheet to help with the calculations.

Exercises 3-6 were designed to be completed using EulersMethodForSystems. The phase portraits in Exercises 5 and 6 are entertaining.

Exercises 7-9 provide some practice at checking solutions, but their main intent is to illustrate the implications of the Uniqueness Theorem. Exercises 12-15 are more abstract versions of the same question.

Exercises 10-11 illustrate how Euler's method for systems applies to second-order equations.

2.5 The Lorenz Equations

If you cover this section, we recommend your mentioning James Gleick's book Chaos. There are also a number of interesting videos that have been produced. They usually do a better job of illustrating the solution curves than we can do with our solvers. One of our favorite videos is Fractals: An Animated Discussion, by Peitgen, Jurgens, Saupe, and Zahlten, published by W. H. Freeman, New York. At this writing, there is also a Java applet written by Patrick Worfolk available at the Geometry Center.

DETools

There are two tools, LorenzEquations and ButterflyEffect, that illustrate the Lorenz attractor.

Comments on selected exercises

Exercises 1-3 and 5 cover aspects of the Lorenz system that can be verified by hand.

Exercise 4 uses ButterflyEffect to illustrate sensitive dependence on initial conditions for the Lorenz attractor.

Review Exercises

Exercises 1-12 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercises 13-24 are true/false problems. We always expect our students to justify their answers. Exercises 19-24 all refer to the same phase portrait.

Exercise 25 involves solving for equilibrium points and interpreting solutions.

Exercises 26-28 involve decoupled or partially decoupled systems.

In Exercises 29-32, students sketch x(t)- and y(t)-graphs from solution curves in the phase plane.

Exercise 33 corresponds to the simple model of a glider that is given in HMSGlider.

Exercise 34 relates to Euler's method for systems.

Comments on the Labs

Lab 2.1 Two Magnets and a Spring

This lab is a computer exploration that involves a bifurcation. It can be started as soon as Section 2.2 is covered, and it pairs up nicely with Lab 4.1.

Lab 2.2 Cooperative and Competitive Species Population Models

This lab can be started as soon as Section 2.2 is covered. It is mostly a computer exploration, but item 2 anticipates certain qualitative ideas that are discussed more specifically in later chapters. Particular attention should be paid to the interpretation of the solutions in physical terms.

Lab 2.3 The Harmonic Oscillator with Modified Damping

Section 2.3 must be covered before this lab can be assigned. This lab is designed to give students practice applying the ideas of this chapter to various second-order equations, both linear and nonlinear ones.

Lab 2.4 A Mass-Spring System with a Rubber Band

This lab is a modification of the standard mass-spring model. It involves both a spring and rubber band. It anticipates the discussion of the Tacoma Narrows Bridge in Chapter 4 and pairs up nicely with Lab 4.3.

Lab 2.5 Active Shock Absorbers

Recently developed magnetorheological fluids allow the damping coefficient of a harmonic oscillator to be adjusted in real time, and this lab examines various models where the damping coefficient is a function of the velocity of the system.