• to give a balanced presentation of analytic, qualitative, and numerical techniques for studying solutions of ordinary differential equations and
• to make the applications of the subject as transparent as possible.

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 approaches in the context of first-order equations. Most of the topics in this chapter can be found in many other texts. The difference lies in the order and emphasis we place on the topics.

Chapter 1 divides naturally into two parts. Sections 1.1-1.5 constitute an 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, and not every section is essential. Section 1.6 (phase lines) anticipates the discussion of the phase plane in Chapter 2, and Section 1.8 (linear equations) is also very important. Students tend to enjoy the models discussed in Section 1.7 (bifurcations), but this section can be skipped if necessary. Also, integrating factors for linear equations (Section 1.9) are not used in subsequent chapters.

At some schools with an engineering-oriented course, 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.

1.1 Modeling via Differential Equations

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 and 2 are straightforward exercises involving exponential growth.

Exercises 3-5 begin the process of thinking qualitatively about differential equations.

In Exercise 4, 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 7-9 involve a simple learning model (see also Lab 1.1).

Exercises 10-14 deal with radioactive decay. Many students already know the formula for the solutions and some wonder why bother with the differential equation. Exercise 13 points out the importance of the differential equation.

Exercises 15 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 15 is to subtract 100t instead of 100.

Exercise 17 is very challenging for students, perhaps because there are multiple "correct" answers, i.e., reasonable models that match the assumptions.

Exercises 19-22 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.

1.2 Analytic Technique: Separation of Variables

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 is 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. FirstOrderExamples also does a good job of illustrating the concept of a general solution, and five of its equations are separable.

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 and 3, students are asked to construct a differential equation given a solution. This change in point of view is difficult for many students.

In Exercise 4, the students solve dP/dt=kP by separating variables.

Exercises 5-32 are standard examples of separable differential equations. However, the algebra necessary to solve for y as a function of t in Exercises 19 and 22 is impossible. The answers to these problems must be left in implicit form. All of the integrals in Exercises 5-32 have closed-form expressions. However, Exercise 22 is challenging.

Exercises 33-39 are fairly standard (and difficult) "word problems." However, Exercise 36 is difficult because setting up a mixing problem in terms of concentration is tricky. In Exercise 35, 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.

1.3 Qualitative Technique: Slope Fields

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 omit 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 Slopes" 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. FirstOrderExamples and TargetPractice are also useful for illustrating slope fields.

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.

1.4 Numerical Technique: Euler's Method

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 1-10 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 5-10 must be done some other way. We have had very good luck using a spreadsheet for these computations. The step sizes in Exercises 1-8 are very large so that only a few computations must be done.

In Exercises 11-13, 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 14 and 15 discuss the geometry of Euler's method. Comparing approximate solutions with the slope field is encouraged.

In Exercises 16-19, the appropriate step size must be determined via experimentation.

Exercises 20 and 21 are applications of Euler's method to root finding. They preview the discussion of phase lines (see Section 1.6).

1.5 Existence and Uniqueness of Solutions

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. Also, the equation dy/dt=y/t + t\cos t in FirstOrderExamples is a good example for the discussion of uniqueness.

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.

1.6 Equilibria and the Phase Line

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. The two autonomous equations in FirstOrderExamples are also useful early in the discussion of this material.

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 29-36 concern the drawing of phase lines using only qualitative information regarding the right-hand side of the differential equation. In Exercises 33-36, the graph of the right-hand side of the equation is sketched from the phase line.

In Exercise 37, students match phase lines with equations from a list. We always insist on valid justifications for their choices.

Exercise 38 concerns possible types of phase lines. Even though this exercise is quite abstract, students usually understand the ideas easily.

Exercises 39 and 40 also involve the drawing of phase lines using only qualitative information regarding the right-hand side of the differential equation.

Exercises 41 and 42 use the PhaseLine tool to anticipate the discussion of one-parameter families of differential equations in Section 1.7.

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.

1.7 Bifurcations

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 20 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 1-8 consider one-parameter families. The students are asked to find the bifurcation values.

Exercises 9 and 10 ask for bifurcation values based on graphs for the right-hand side of the differential equation.

In Exercises 11 and 12, students must provide graphs that are consistent with the bifurcations described.

Exercises 13 and 14 are more abstract. They ask if it is possible to find a one-parameter family of a certain type with specified bifurcations.

Exercises 15-21 return to population and harvesting examples. Exercise 20 requires interpretation of bifurcation diagram. The conclusions are quite striking.

Exercises 22 and 23 use the bifurcation plane feature in PhaseLines.

1.8 Linear Equations

In this section, we introduce linear differential equations, and we discuss the special structure of their solution sets as given by the Linearity Principle and the Extended Linearity Principle. This presentation anticipates the treatment of linear systems in Chapter 3 and second-order linear equations in Chapter 4. We also discuss the Method of Undetermined Coefficients for constant-coefficient equations rather than integrating factors. (Integrating factors are discussed in the next section.) This approach pays dividends later and is also consistent with the what our colleagues in engineering do in their courses.

DETools

HPGSolver can be used to illustrate the Linearity Principle and the Extended Linearity Principle.

Comments on selected exercises

Exercises 1-12 provide routine practice finding general solutions and solutions to initial-value problems.

In Exercises 13 and 14, students are asked to explain why the method works the way that it does.

Exercises 15 and 16 involve graphical illustrations of the two linearity principles.

Exercises 17 and 18 involve theoretical aspects of the Linearity Principle.

Exercises 19-24 illustrate how we deal with forcing functions that are sums of the elementary examples.

Exercises 25-28 are qualitative in nature. In fact, they are a little tricky. If you assign them, you may want to hint to your students that they should visualize the slope fields.

Exercises 29-31 are investment problems.

In Exercise 32, students are asked to verify an assertion made in the section. This exercise is not difficult if they remember the easy way to check a possible solution.

Exercise 33 leads the students through a verification of the Extended Linearity Principle. This verification was left to the reader in the section.

Exercise 34 is a tricky theoretical exercise involving the Linearity Principle.

1.9 Integrating Factors for Linear Equations

For many instructors, using integrating factors to solve first-order linear equations is a necessity. However, others feel that the approach to linear equations that we take in Section 1.8 is sufficient. Consequently, we end this chapter with a discussion of integrating factors and leave the decision to cover this technique to the instructor. Some prefer to start Chapter 2 as quickly as possible while others think that it is good for the students to see another analytic technique at this point in the course. Integrating factors are mentioned two or three times throughout the rest of the text, but they are never used to solve an equation. Therefore, this section can be omitted without serious consequences.

Our presentation in 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 that come from even relatively simple linear equations.

Comments on selected exercises

Exercises 1-12 are standard. In some cases, the integration is challenging.

Exercises 13-20 consider the possibility of encountering impossible integrals when solving linear equations. They point out how often this occurs.

Exercise 21 asks the student to compare the method of integrating factors with the guessing technique described in Section 1.8.

Exercise 22 is a theoretical exercise that relates the abstract formula for the general solution obtained in this section with the Extended Linearity Principle from Section 1.8.

Exercise 23 uses integrating factors to justify the "second" guess that we introduced in Section 1.8.

Exercises 24-27 are mixing problems. Exercise 27 considers the extreme case when the initial volume is zero (division by zero is a danger).

Review Exercises

Exercises 1-10 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercises 11-20 are true/false problems. We always expect our students to justify their answers.

Exercises 21-40 are routine exercises where students must derive general solutions or solve initial-value problems. Exercises 26, 32, and 39 involve integrating factors.

Exercise 41 involves Euler's method and a solution that blows up in finite time.

Exercises 42-45 involve qualitative analysis in one form or another.

Exercise 46 investigates an equation that is both linear and separable.

Exercise 47 concerns the Uniqueness Theorem.

Exercise 48 involves Euler's method and phase lines.

Exercise 49 uses Newton's law of cooling

Exercises 50-51 are bifurcation problems.

Exercise 52 involves compound interest.

Exercise 53 investigates the phase line of a simple linear equation.

Exercise 54 is a modeling exercise based on the number of cell phone subscribers in the U.S.

Exercise 55 examines solutions that are defined for all time for a particular equation.

Exercises 56 and 57 are mixing problems.

Comments on the Labs

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 Oil Production

This lab involves U.S. and world oil production from 1920-2000. The students are asked to use the logistic to model oil production and then analyze their results.