Section 2.1 introduces the terminology and ideas via a natural predator-prey example. In Sections 2.2 and 2.3, the notions of autonomous systems, vector fields, direction fields, phase planes, solutions and equilibrium points are presented. These sections are closely related and can be thought of as one long section which requires a couple of classes to cover. Linear, homogeneous, constant coefficient, second-order equations (harmonic oscillators) are introduced and related to systems in Section 2.4. Euler's method for first-order systems and second-order equations is covered in Section 2.5, and in Section 2.6 we discuss qualitative methods for drawing phase planes. Section 2.7 begins the qualitative discussion of the Lorenz system as an introduction to three-dimensional systems.

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 The Predator-Prey Model

Using the analysis of a single predator-prey model, the basic ideas of first-order systems are introduced in this section. Our goal is to introduce the relationship between various graphical representations of the system, its solutions, and the interpretation of the solutions in terms of the model. The graphical representations include the phase plane, the graphs of the component functions of the solutions, the vector field, and the direction field.

The transition to systems is natural for students. Relating the solution in the phase plane to the graphs of the components is considered difficult by some, but is generally mastered after sufficient effort.

Some instructors feel that too much new material is presented in this section. However, we repeat the basic ideas in a more general setting in Sections 2.2 and 2.3, and we cover Section 2.1 quickly. We like to use the material of Section 2.1 as a running example for the more formal definitions in the next two sections.

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. This is easier than developing models from scratch, but it is still challenging. In Exercise 18, there is more than one reasonable answer.

In Exercises 19--24, models for concentrations of reactants in simple chemical reactions are developed. These models reappear in subsequent sections (Section 2.3, Exercises 22--26, and Section 2.6, Exercises 14--18).

2.2 Systems of Differential Equations

This section establishes notation and terminology for systems. Vectors are introduced along with many adjectives for describing systems. Vector fields, direction fields, and equilibrium points are also discussed.

One major goal of the section is to develop an understanding of what initial conditions and solutions are and how to check (by substituting into the system) that a given vector-valued function is a solution. Once students master the ability to check solutions of systems (and that they come to think of the idea as natural), they have reached an important mile post in their understanding of systems.

Comments on selected exercises

Exercises 1--8 concern the vocabulary for systems. In Exercises 36 and 37, the relationship between the systems in Exercises 5 and 7 and the systems in Exercises 6 and 8 (respectively) are explored.

Exercises 9--16 and 21--27 involve checking that given functions are solutions of a given system. This task is straightforward but extremely necessary.

In Exercises 17--20, direction fields are matched to systems. This type of exercise is less tedious than sketching direction fields by hand. If essays are required to justify why a given system corresponds to a particular direction field, students must examine the fields closely.

Exercises 28--35 ask for equilibrium points and sketches of direction fields for given systems. These systems reappear in Section 2.3, Exercises 1--8, where the phase plane is also requested.

2.3 Graphical Representation of Solutions of Systems

In this section we look at the various graphs of solutions of systems and how sketches of these graphs can be generated from the direction field. The relationship between a solution curve in the phase plane and the graphs of the component functions is difficult at first, but it is eventually mastered by most students. 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 2.26).

Some examples are given for which formulas for solutions can be found, and the idea of the general solution of a system is briefly introduced. Also, the Existence-Uniqueness Theorem for system is briefly stated. As with first-order equations, it is the uniqueness half of the theorem which is stressed since it is the most useful for drawing phase planes of autonomous systems.

Comments on selected exercises

Exercises 1--8 ask for a detailed analysis of the given complicated systems. Equilibrium points can be found by hand. Ideally, technology should be used to draw the direction fields, then the students should sketch the solution curves on top of these fields.

The systems in Exercises 9--12 can be explicitly solved because the systems decouple. These exercises are a good review of separable and linear equations as discussed in Chapter 1.

The systems in Exercises 13--16 can be explicitly solved for the given initial condition due to some special geometry of the system. These also give a good review of separable and linear equations and are quite difficult.

Exercises 17--21 concern a model of an arms race. Use of technology for drawing direction fields and phase planes should be encouraged, e.g., to approximate the coordinates of the equilibrium points.

Exercises 22--26 begin the analysis of the chemical reaction models introduced in the exercise set in Section 2.1. These models reappear in Section 2.6, Exercises 14--18.

Exercises 27--31 concern the Uniqueness Theorem for systems.

Exercise 32 gives an example of a solution that is not defined for all real numbers.

2.4 Second Order Equations and the Harmonic Oscillator

In this section, we derive the second-order equation for the motion of a harmonic oscillator using Newton's and Hooke's laws. This second-order equation is then converted into a first-order system, and we analyze examples using the vector field. We present solution techniques in detail in Chapter 3.

We relate the qualitative description of solutions to the what is physically reasonable for the mass-spring harmonic oscillator. This approach is both important and dangerous. Students sometimes think that the physical argument \textit{is} the analysis of the system rather than simply a check of the results of the mathematical analysis.

We chose not to include the standard ``guess-and-test'' method for solving the harmonic oscillator equation at this point for several reasons. First, we wish to maintain the emphasis on the qualitative analysis of solutions. Second, since we have not discussed the significance of linearity, it is difficult to do much with guess-and-test at this point. We return to this discussion in Section 3.1 after the Linearity Principle is discussed.

It is certainly possible to skip directly to Chapter 3 at this point, but we prefer to cover the material (with the possible exception of Section 2.7) in the order given in order to maintain the balance among the analytic, numeric, and qualitative approaches.

Comments on selected exercises

Exercises 1--4 involve the conversion of second-, third-, and fourth-order equations into first-order systems.

Exercises 5--7 derive the equations for a hanging spring with gravity as extra force.

Exercises 8--11 ask for the direction field and qualitative behavior of solutions for the harmonic oscillator with given coefficients. We encourage the use of technology in these exercises.

Exercises 12--14 are similar to Exercises 5--7. A system involving two opposing springs is considered.

Exercise 15 asks for the model of hard and soft springs (see also Lab 4.1).

Exercises 16--20 concern a model for a flexible suspension bridge. This model is covered in detail in Section 5.4, and these problems are quite challenging at this point in the course.

2.5 Euler's Method for Autonomous Systems

The approach to Euler's method is kept as simple and geometric as possible. Again, the most difficult point is the relationship between the graphs. For example, solutions near an equilibrium point move slowly (the vectors in the vector field are small), so the Euler's approximation in the phase plane is made up of small steps.

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--11 refer to the swaying building model. One of two models is to be selected by comparing to given numerical data. Exercise 11 asks what experiment should be done to distinguish between the two systems.

2.6 Qualitative Analysis

In this section we use the direction field, along with some numerics when necessary, to study the long-term behavior of solutions of nonlinear systems. The only new technique introduced is the location of nullclines in the phase plane. Unfortunately, many students are confused initially about the difference between nullclines and solution curves.

Geometric analysis of this sort is particularly hard for students because is involves many steps and many different ideas and techniques. (They keep hoping you will just give them the magic bullet for understanding systems and are skeptical when you say there isn't one.) Extended projects are particularly helpful in making students realize that there is no template that leads to a complete of a phase plane.

Comments on selected exercises

In Exercises 1--6, 10--13, and 14--18, a qualitative analysis of the given system is requested. This analysis should go beyond what a student can print out from a good numerical solver. Exercises 14--18 relate to the chemical reaction systems of Section 2.1 (Exercises 19--24) and Section 2.3 (Exercises 22--26).

Exercise 7 is a fairly hard problem on geometry of solutions in the phase plane.

Exercises 8 and 9 concern the general Volterra-Lotka models of a pair of species.

Exercises 19--21 study a nonlinear saddle.

2.7 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 3.7, and the Lorenz system is studied more carefully in Sections 4.4 and 6.4.

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.

Comments on selected exercises

Exercises 1--5 cover details of the Lorenz system that can be easily verified by hand

Exercise 6 requires some fairly sophisticated numerics to compare solutions of the Lorenz system.

Comments on the Labs

All 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: Cooperative and Competitive Species Population Models

This lab can be started as soon as Section 2.1 is covered. It can either be a purely computer exploration, or if Section 2.6 has been covered, it can include some more careful qualitative analysis. Particular attention should be paid to the interpretation of the solutions in physical terms.

Lab 2.2: The Harmonic Oscillator with Modified Damping

Section 2.4 must be covered before this lab can be assigned. The first part relates to the harmonic oscillator. Hence, that portion of the lab should be complete before going far into Chapter 3.

Lab 2.3: Swaying Building Models

This lab requires a solver that produces numeric data (rather than just graphs). It can be done with a programmable calculator.

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Version 1.1. May, 1996.