To accomplish these goals, we continually introduce new models and modify old ones using all three types of techniques whenever possible. In Chapter 1 we introduce, by example, all of these themes for first-order equations. Most of the topics in this chapter can be found in many other texts. The difference lies in the order and the emphasis we place on the topics.
Chapter 1 divides naturally into two parts. Sections 1.1-1.5 constitute the introduction to modeling and to examples of analytic, qualitative, and numerical techniques along with the Existence and the Uniqueness Theorems (and their applications). These sections are essential both in material and in outlook to the rest of the text.
Sections 1.6-1.9 discuss more specialized topics. It is possible to cover selected sections. In any case, Section 1.6 (phase lines) is very important to the development of systems in Chapter 2, and Section 1.8 (linear equations) is probably too standard to be skipped. (Note our comments related to Appendix A in the following discussion of Section 1.8.)
At some schools with a particularly engineering-oriented program, an early introduction to Laplace transforms is desirable. Consequently, we have written Sections 6.1 and 6.2 so that they can be covered immediately after Chapter 1. However, in our experience, it is best to wait until Chapter 4 has been covered before discussing Laplace transforms.
Our main goal in this section is to demonstrate that differential equations arise naturally in "real world" systems and that solutions should be interpreted in terms of the original system. In building models, we emphasize that there is a connection between the assumptions made regarding the physical system and the individual terms in the differential equation. In studying solutions, we stress that the formula and/or the graph of a solution provides a prediction about the future behavior of the physical system under consideration. We also emphasize that models are imperfect and should be criticized and improved if their predictions do not correspond to reality.
DETools
Using HPGSolver to graph solutions to the logistic equation is a good way to introduce students to the use of technology in this course, but keep the slope field turned off until Section 1.3.
Comments on selected exercises
Exercises 1-3 begin the process of thinking qualitatively about differential equations.
In Exercises 4 and 17, students are asked to model data with an exponential model. However, exponential growth gives a poor fit to the data as well as ridiculous predictions of the future population. If pressed, students will suggest a logistic model as an alternative.
Exercises 5-7 deal with a simple learning model (see also Lab 1.1).
Exercises 8-12 deal with radioactive decay. Many students already know the formula for the solutions and some wonder why bother with the differential equation. Exercise 11 points out the importance of the differential equation.
Exercises 13, 14, and 16 are "do-able" modeling problems. In general, students find modifying existing models easier than creating new ones from scratch. A standard mistake in part (a) of Exercise 13 is to subtract 100t instead of 100.
Exercise 15 is very challenging for students, perhaps because there are multiple "correct" answers, i.e., reasonable models that match the assumptions.
Exercises 18-21 deal with systems of differential equations (population models with two populations), but only in terms of understanding the relationship between terms in the equations and assumptions about the physical system.
In this section we give the first examples of the "analytic" approach (i.e., finding closed-form formulas for solutions) to differential equations. We begin by emphasizing that solutions of differential equations can be checked by substitution into the differential equation. Understanding this fact is fundamental to learning the relationship between a differential equation and its solutions, but it is a fact that students often forget later in the course. Repeated reference to this fact throughout the course is important.
The only equations that we consider at this point are separable, and we include the standard mixing and compound interest examples. The only nonstandard aspect of this section is the observation that, even if an equation is separable, there is no guarantee that the solution is obtainable in closed-form since either the integrals or the algebra may be impossible. The mixing example returns in Section 1.3 where it an analyzed qualitatively.
Students find this section fairly straightforward and familiar. Although the students may need some review of their techniques of integration, the only real difficulty is in dealing with absolute value signs. Most textbooks ignore this technical detail, and there is merit to that approach.
DETools
HPGSolver can be used to illustrate the solution to an initial-value problem as well as the general solution of a differential equation.
Comments on selected exercises
Knowing what it means to check a solution by "plugging it back into" the differential equation is an excellent way to reinforce what a solution is. Exercise 1 illustrates this fact, and this check is requested in many other problems throughout the book.
In Exercises 2-4, students are asked to construct a differential equation given a solution. This is a change in point of view that many find difficult.
Exercises 5-34 are standard examples of separable differential equations. However, the algebra necessary to solve for y as a function of t in Exercises 19, 23 and 24 is impossible. In Exercise 14, solving for y is difficult because one must solve a cubic. The answers to these problems must be left in implicit form. All of the integrals in Exercises 5-34 have closed-form expressions. However, Exercise 24 is challenging.
Exercises 35-40 are fairly standard (and difficult) "word problems." However, Exercise 38 is difficult because setting up a mixing problem in terms of concentration is tricky. In Exercise 41, the first step is to determine the proportionality constant for Newton's law of cooling. Wherever possible, we ask for a description of the long-term behavior of a solution to promote qualitative thinking.
In this section we introduce our first geometric and qualitative technique---slope fields. We emphasize that the slope field is a tool to help in sketching the graphs of solutions. This point helps drive home the idea that solutions are functions. We also encourage describing the behavior of solutions qualitatively. The discussion of slope fields leads naturally to Euler's method as well as issues of uniqueness of solutions.
We introduce RC circuits as an example in this section, but we do not develop the physics or the circuit theory. Many, but not all, of our students concurrently take electric circuit theory. For these students, developing the theory of RC circuits is review (or preview). For students who are not studying engineering, the topic is entirely new. Hence, we forego the theory and start with the differential equation.
DETools
It is extremely helpful at this point to have some sort of technology available for drawing slope fields, e.g., HPGSolver. Start with the ``Draw Slope Mark'' box checked and place individual slope marks at random places in the ty-plane. Then turn on the slope field, and illustrate the relationship between the slope field and graphs of solutions.
Comments on selected exercises
Exercises 1-6 are very easy with technology, but it is worthwhile to encourage students to do them to gain intuition about what slope fields look like and to practice sketching solutions.
Exercises 7-10 involve practice going from slope field pictures to graphs of solutions.
In Exercise 15, since technology can be used to determine which slope field is which and that is not the point of the exercise, students should be encouraged (required) to write a paragraph that describes what features of the slope fields they used to determine which is which.
Exercises 11, 13, 14, 16, and 17 are somewhat more challenging theoretical problems relating slope fields and solutions.
Exercise 18(c) is a good excuse to start worrying about the uniqueness of solutions.
Exercises 19-23 involve the RC circuit model. They serve as a good advertisement for qualitative and numeric methods.
Exercise 24 involves a discontinuous differential equation (a population model where a disease is introduced at time t=5). Again, qualitative and numeric methods are best. HPGSolver has a step function to handle equations of this sort. Assigning only parts (a), (b), and (d), avoiding finding solutions in closed form, is a reasonable modification of this problem.
In this section we introduce Euler's method. This topic naturally follows slope fields (as a way to have the computer sketch accurately what we can sketch roughly by hand from the slope field). Introducing a numerical method early, even one as elementary as Euler's method, is important to legitimize the use of the technology which is essential for the labs and for the development of qualitative techniques in our approach.
A careful analysis of error in Euler's method is given in Section 7.1, and that section can be covered at this point in the course. Improved Euler's method and Runge-Kutta are also presented in Chapter 7 (see the section-by-section comments for Chapter 7).
We have tried to promote a healthy skepticism---a middle ground between blind faith and paranoia---for numerical methods. Numerical methods usually do work, and in many cases they are the technique of choice even when more theoretical approaches are available. The importance of interpretation must be stressed.
DETools
EulersMethod is a tool that illustrates Euler's method for a variety of equations. It can be used in class when Euler's method is first introduced, and Exercises 1-4 refer to it specifically.
Comments on selected exercises
Exercises 1-8 are computational practice with Euler's method. The first four refer to EulersMethod but could be done without using the tool. The computations in Exercises 5-8 must be done some other way. We have had very good luck using a spreadsheet for these computations. All of the step sizes are very large so that only a few computations must be done.
In Exercises 9-11, the inaccuracies of "large step" Euler's method are discussed. This is another good opportunity to begin speculation on the Uniqueness Theorem. (How do we know solutions don't jump over equilibrium solutions?)
Exercises 12 and 13 discuss the geometry of Euler's method. Comparing approximate solutions with the slope field is encouraged.
In Exercises 14-17, the appropriate step size must be determined via experimentation.
Exercises 18 and 19 are applications of Euler's method to root finding. They preview the discussion of phase lines (see Section 1.6).
In this section we present the existence and uniqueness theory for first-order equations. We also consider the issue of the domain of definition of a solution---a theoretical point that has been ignored in Sections 1.1-1.4.
Existence is stated and then taken for granted. Uniqueness, however, is emphasized as a very useful tool for the qualitative study of solutions. This is a good opportunity to show that "abstract theorems" are actually quite useful in applied mathematics.
We also deal with questions of the domain of definition for solutions. The exceptional student will have wondered about domains of definition in Section 1.2 where restricted domains are the norm. We take the dynamical systems point of view that a solution that escapes to infinity or that encounters a singular point of the differential equation at a finite time cannot be extended beyond that time. For example, the function y(t)=1/t for t>0 is a completely different solution to the equation dy/dt = -y2 than the solution y(t)=1/t for t<0.
DETools
HPGSolver is useful both during class and on the homework.
Comments on selected exercises
Exercises 1-9 encourage the use of the Uniqueness Theorem to take a small amount of information about solutions of a differential equation and derive information about other solutions. The information thus gained is qualitative.
Exercises 10 and 18 both deal with the situation where the Uniqueness Theorem does not hold. In general we try not to make too big a deal of lack of uniqueness. However, for some of our students, this is the first time they have used a theoretical result in a practical way and they need to remember that theorems have hypotheses.
Exercise 11 (showing that solutions of autonomous equations cannot have local maxima and minima) is difficult.
Exercises 12-17 are exercises that address the domain of definition issue. They are also a good review of Section 1.2.
This section deals exclusively with autonomous equations. It introduces the notion of a phase line. The goal is to promote qualitative analysis. The section also sets the stage for the concept of the phase plane, which will be introduced in Chapter 2.
At this point, students sometimes feel overwhelmed with graphs. For a single equation of the form dy/dt = f(y), they have the graph of f(y), the slope field, the graphs of solutions, and the phase line. We try to emphasize that this is good---the more ways of representing a differential equation we have, the more ways we have of obtaining information about its solutions.
The section is long and difficult to cover in a single one-hour lecture. The material at the end of the section on linearization (pp. 87-89) can be skipped with no serious ramifications. This material is included as a preview of the topic of linearization as it is discussed in Chapter 5, but that discussion is a long way off.
DETools
PhaseLines can be used to illustrate the relationships among the graphs of solutions, the phase line, and the graph of the right-hand side of the differential equation. At this point you probably want to keep the bifurcation plane hidden.
Comments on selected exercises
Exercises 1-28 involve drawing phase lines and using them to obtain sketches and other qualitative information about solutions. The role of the Uniqueness Theorem can be emphasized.
Exercises 29-38 concern the drawing of phase lines using only qualitative information regarding the right-hand side of the differential equation. In Exercises 33-38, the graph of the right-hand side of the equation is sketched from the phase line.
Exercises 39 and 40 use the PhaseLine tool to anticipate the discussion of one-parameter families of differential equations in Section 1.7.
Exercises 41 and 42 concern possible types of phase lines. Even though these exercises are quite abstract, students usually understand the ideas easily.
Exercise 43 concerns the behavior of solutions near equilbria in cases where linearization is not conclusive.
Exercise 44 compares solutions for an equation with singularities on the phase line.
Exercises 45-48 can be applied to any transit system where busses and trains run frequently without a rigid schedule. The interpretation is difficult but, from painful experiment, fairly accurate.
Highlighting the role and importance of parameters in the study of differential equations is one of our major goals. Using phase lines, parameters can be made quite geometric and accessible. This section also reinforces the notion of a phase line.
Simply having some familarity with the term "bifurcation" is very useful in subsequent sections. For example, an understanding of node equilibrium points and bifurcations on phase lines helps a great deal when discussing critically damped oscillators and linear systems that have 0 as an eigenvalue. If time permits, the bifurcation diagram can also be discussed.
Some of us were initially hesitant to tackle this subject at this level. However, after doing so, we became true believers. Exercise 12 illustrates the power of the ideas introduced in this section.
Some of us view this section as central to our approach while others consider it to be an interesting optional section. In the latter case, this section is skipped during "short" semesters.
DETools
This section is the appropriate place to turn on the bifurcation plane in PhaseLines.
Comments on selected exercises
Exercises 1-6 consider one-parameter families. The students are asked to find the bifurcation values.
Exercises 7-14 return to population and harvesting examples. Exercise 12 requires interpretation of bifurcation diagram. The conclusions are quite striking.
Exercises 15 and 16 use the bifurcation plane feature in PhaseLines.
Exercises 17-19 are more theoretical, using the index of an equilibrium point as introduced in Exercise 42 in Section 1.6.
Exercises 20-24 involve some unusual bifurcations.
Linear (nonautonomous) equations occur in sufficiently many of the standard examples that it seems appropriate to introduce the technique of integrating factors. This section is fairly standard, although we have made an attempt to motivate the ideas behind integrating factors as much as possible. We have also tried to be honest about the difficulties that frequently arise in the integrals encountered with even relatively unimposing linear equations.
Recently based on discussions with our colleagues in engineering, some of us have replaced the topic of integrating factors with the Method of Undetermined Coefficients for constant-coefficient, first-order equations. This alternate approach is presented in Appendix A. This appendix can be used as a replacement or a supplement to this section. In any case, the students should be familiar with the terminology introduced on pages 113 and 114.
Comments on selected exercises
Exercises 1-14 are standard. In some cases, the integration is challenging.
Exercises 15--22 consider the possibility of encountering impossible integrals when solving linear equations. They point out how often this occurs.
Exercises 23-25 are investment problems, and Exercises 26-29 are mixing problems. Exercise 29 considers the extreme case when the initial volume is zero (division by zero is a danger).
Many of the standard analytic techniques and many of the special cases for analytically solving differential equations can be viewed as a "change of variables." While this section is optional, it does provide reinforcement and review of slope fields, phase lines, and analytic techniques. It also introduces the geometry associated to a change of variables. (Another simple change of variables occurs in Section 5.1.)
Only the dependent variable is changed in the text and exercises (with the exception of Exercise 26).
Comments on selected exercises
Exercises 1-13 relate solutions of equations in different variables, both analytically and geometrically.
Exercises 14-16 apply this technique to mixing problems (changing the dependent variable to concentration).
Exercises 17-25 illustrate changes of variables to help in "linearization" near equilibrium points.
In Exercise 26, changing the independent variable is discussed.
The labs are written with the expectation that the students will use a numerical solver such as HPGSolver. It is surprising how reluctant some students are to use technology.
Lab 1.1 Rate of Memorization Model
Because the model equation can be solved easily, this lab can be done without technology. (In fact, most students chose to solve the equation analytically and fit the data to the solution rather than take advantage of numerical techniques.) There is considerable variability in the results of the experiment. This lab can be started immediately after Section 1.1.
Lab 1.2 Growth of a Population of Mold
While this lab is a lot of fun, it is also subject to many difficulties. Some pieces of bread just dry out, leaving the student with no data. Also, be prepared for all sorts of strange stories why the report is not done on time---my dog ate my homework might actually be true. It seems that, on most pieces of bread, the mold stops growing before the entire piece is covered, i.e., the carrying capacity is less than 1. This lab can be started immediately after Section 1.1, and it takes at least two weeks to collect a reasonable amount of mold.
Lab 1.3 Logistic Population Models with Harvesting
The equations in this lab are difficult to analyze theoretically. Hence, the use of a solver is a must. This lab relates most closely with Section 1.7, but it can be done even if the material in Section 1.7 is not discussed in class. It has proven to be one of our most successful.
Lab 1.4 Exponential and Logistic Population Models
This lab points out the difficulties in making generalizations while modeling. There is considerable variability in the behavior of the populations of the states. The process of accepting or rejecting and adapting a model is new to many, if not most, students. Also, determining parameter values from the data is an important concept that many have not seen before. This lab can be started immediately after Section 1.1 is covered.
Lab 1.5 Modeling the Extinction of the Passenger Pigeon
This lab involves the fascinating story of the extinction of the passenger pigeon. It begins with an analysis of a logistic model with constant harvesting. Ultimately students are asked to investigate the behavior of a more complicated model for various parameter values.