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. Then 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-dimen\-sional 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. There 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 availability of some sort of technology that students can use to draw vector fields, direction fields, and phase planes is essential. In order to begin to get a feel for what pictures to expect, students must see many examples accurately drawn. Software is available from several sources (check our web page for more specific information).
2.1 Modeling Via Systems
This modeling section is similar in purpose to Section 1.1. Our goal is to introduce the relationship between various graphical representations of a system of differential equations, its solutions, and the interpretation of the solutions in terms of the model. The graphical representations include the phase plane and graphs of the component functions of the solutions.
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.
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 that 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 19--24, 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
This section introduces geometric and qualitative ideas that we use throughout the remainder of the text. Vector fields, direction fields, solution curves in the phase plane, equilibrium solutions, and phase portraits are discussed. Vector fields for autonomous systems play the role of slope fields for first-order equations. They are the fundamental geometric representation of systems of differential equations. We find the software tools contained in Interactive Differential Equations (IDE) designed by Hubert Hohn with the assistance of authors Beverly West, Steven Strogatz, Jean Marie McDill, and John Cantwell to be extremely useful at this point in the course. Many of the tools in IDE do an excellent job of driving home the concepts introduced in this section. (IDE is published by Addison-Wesley.)
Comments on selected exercises
Exercises 1--6 involve vector fields and phase portraits for very simple systems. These exercises are designed to be done without the use of technology.
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.
In Exercises 11--16, students are expected to solve for the equilibrium points by hand, but then they are expected to 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.
Exercises 17--20 taken together form 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 essays are required to justify why a given system corresponds to a particular direction field, students must examine the fields closely. Exercises 21--24 match solution curves with their corresponding x(t)- and y(t)-graphs.
In Exercises 25--28, 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 29 goes the opposite direction as Exercises 25--28. 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 30 and 31 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
Many students are uncomfortable with the heavy emphasis on geometry in Sections 2.1 and 2.2. Consequently this section is devoted to certain special systems for which closed-form solutions can be obtained. This is also a convenient place to repeat the observation that any given function can always be checked to see if it is a solution, and we begin this section with that observation.
Using integrating factors from Section 1.8 or undetermined coefficients from Appendix A, 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 178 and 179 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) = est. Since the Linearity Principle does not appear until Section 3.1, we stop short of finding general solutions of the form y(t) = k1 es1 t + k2 es2 t. However, we do make the connection between solutions of the form y(t) = est and the corresponding vector-valued solution Y(t) = (y(t), v(t)) = (est, s est).
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--16 provide the same type of practice for second-order equations. These equations reappear in Exercises 21--24 of Section 3.2.
Exercises 17--18 involve the analysis of a mass-spring system that involves two springs.
2.4 Euler's Method for Autonomous Systems
Section 2.4 generalizes Euler's method to systems, and we are now able to explain how the graphs and solution curves for the predator-prey equations in Sections 2.1 and 2.2 are obtained. Students usually have little difficulty understanding Euler's method for systems if they have a solid understanding of Section 1.4 and the vector field ideas introduced in Section 2.2.
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.
Comments on selected exercises
Exercises 1--6 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.
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
We introduce the Lorenz system here mainly because it is possible to do so. Almost none of our students have seen any modern (i.e., post 1800) mathematics, and they are surprised to learn that there are unanswered questions and that there is active research in mathematics. At this point, we can only describe the Lorenz system and display some numerical solutions. Consequently, this is something of a "golly-gee-whiz" section. Three-dimensional linear systems are discussed in Section 3.8, and the Lorenz system is studied more carefully in Sections 5.5 and 8.5.
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 from the Geometry Center http://www.geom.umn.edu/java/Lorenz/).
Comments on selected exercises
Exercises 1--3 and 5 cover details of the Lorenz system that can be verified by hand.
Exercise 4 requires some fairly sophisticated numerics to compare solutions of the Lorenz system.
Comments on the Labs
Both of these labs require technology capable of sketching solutions in the phase plane. The ability to draw graphs of the coordinate functions is also very useful.
Lab 2.1 The Harmonic Oscillator with Modified Damping
Section 2.3 must be covered before this lab can be assigned. This lab is designed to give the students practice applying the ideas of this chapter to various second-order equations, both linear and nonlinear ones.
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.