Chapter 1 divides naturally into two parts. Sections 1.1--1.5 constitute the introduction to modeling and to examples of analytical, qualitative and numerical techniques along with the Existence Uniqueness theorem (and its 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 topics. However, 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.
1.1 Modeling Via Differential Equations
Our main goal in this section is to demonstrate that differential equations arise naturally in physical "real world" systems and that solutions should be interpreted in terms of the physical system. In building models, we emphasize that there is a connection between the assumptions made about 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.
Comments on selected exercises
Exercises 1 to 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 1.3).
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--15 are "do-able" modeling problems. In general, students find modifying models easier than creating them from scratch. A standard mistake in 13(a) is to subtract 100t instead of 100.
Exercise 16 seems to be 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, but only in terms of understanding the relationship between terms in the equations and assumptions about the physical system.
1.2 Analytical Technique: Separation of Variables
In this section we give the first examples of the "analytical" 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 is the first step in learning the relationship between a differential equation and its solutions.
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. While they complain about the partial fractions, the only real difficulty is in dealing with absolute value signs. Most textbooks ignore this technical detail, and there is merit to that approach. 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.
Comments on selected exercises
In Exercises 2--4, students are asked to construct a differential equation given a solution. This is a change in point of view that some 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 very 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.
1.3 Qualitative Technique: Slope Fields
In this section we introduce our first geometric and qualitative technique --- slope fields. It is extremely helpful at this point to have some sort of technology available for drawing slope fields, e.g., a computer or a graphing calculator. Making acceptable pictures by hand is very tedious. 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 (ECT). 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.
In Figure 1.28, you should replace K = 1 with K = 2.
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. The answer to Exercise 1 in the back of the book is wrong. See the errata. You may want to ask the students for a quick reason why the answer in the back of the book is obviously wrong.
Exercises 7--10 involve practice going from slope field pictures to graphs of solutions.
In Exercise 11, technology can be used to determine which slope field is which. Students should be encouraged (required) to write a paragraph that describes the qualitative features of the slope fields that they used to determine which is which (without technology).
Exercises 13--17 are somewhat more challenging theoretical problems relating slope fields and solutions.
Exercise 18(b) is a good excuse to start worrying about the uniqueness of solutions.
Exercises 19--23 involve the RC circuit model and are very tedious. Hence, they serve as a good advertisement for qualitative methods.
Exercise 24 involves a discontinuous term (a population model where a disease is introduced at time t=5). Again, qualitative methods are best. (Assigning only parts (a), (b) and (d), avoiding finding solutions in closed form, is very reasonable).
1.4 Numerical Technique: Euler's Method
In this section we introduce Euler's method. This numerical topic naturally follows slope fields (as a way to have the computer sketch accurately what we can sketch by hand from the slope field picture). 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.
Comments on selected exercises
Exercises 1--8 are computational practice with Euler's method. Doing (some of) these problems by hand or writing a calculator or a computer program should be encouraged. Using a canned program makes these exercises less useful. All the step sizes are very large so that they can be done by hand.
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. This previews the discussion of phase lines (see Section 1.6). Lab 1.4 is very similar.
1.5 Existence and Uniqueness of Solutions
In this section we present the Existence-Uniqueness Theorem. We also consider the issue of the domain of definition of a solution --- a theoretical point which has previously been ignored.
Existence is stated and then taken for granted. Uniqueness, however, is emphasized as a very useful tool for the qualitative study of solutions. We take this as 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 from y(t)=1/t for t<0.
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 a more 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. Both concern the differential equation dy/dt=y^{2/3}. We try not to make too big a deal of this. 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 also make 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 and dynamical analysis. The section also sets the stage for the concept of the phase plane that 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 solution curves, and (now) 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. 81--83) can be skipped with no serious ramifications. That material is included as a preview of the topic of linearization as it is discussed in Chapter 4, but that discussion is a long way off.
Comments on selected Exercises
Exercises 1--22 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 23--32 concern the drawing of phase lines using only qualitative information about the right-hand side of the differential equation. In Exercises 27--32, the graph of the right-hand side of the equation is sketched from the phase line.
Exercises 33 and 34 concern possible types of phase lines. Even though these exercises are quite abstract, students usually understand the ideas easily.
Exercise 35 concerns the behavior of solutions near equilbria when linearization is not conclusive.
Exercise 36 compares solutions for an equation with singularities with the phase line.
Exercises 37--40 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 geometrical 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 10 illustrates the power of the ideas introduced in this section.
Comments on selected exercises
Exercises 1--9 and 11--12 consider simple one- and two-parameter families, respectively. The students are asked to find the bifurcation values. This material is especially suited to the use of graphing technology.
Exercises 7--10 return to population and harvesting examples. Exercise 10 relys on interpretation of bifurcation diagram. The conclusions are quite striking.
Exercises 13--15 are more theoretical, using the index of an equilibrium point as introduced in Exercise 34 in Section 1.6.
Exercises 16--20 display some unusual bifurcations. Again, this is a good place for graphing technology.
1.8 Linear Differential Equations
Linear (nonautonomous) equations occur in sufficiently many of the standard examples that it seems appropriate to introduce 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.
Comments on selected exercises
Exercises 1--18 are standard. In some cases, the integration is challenging.
Exercises 19 and 20 consider the problem of encountering impossible integrals when solving linear equations, pointing out how commonly this occurs.
Exercises 21--23 are investment problems, and Exercises 24--27 are mixing problems. Exercise 27 considers the extreme case when the initial volume is zero (division by zero is a danger).
1.9 Changing Variables
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 can be safely skipped, it does provide a reinforcement and review of slope fields, phase lines and analytical techniques. It also introduces the geometry associated to a change of variables. (Another simple change of variables occurs in Section 4.1.)
Only the dependent variable is changed in the text and exercises (with the exception of Exercises 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). The answer in the back of the book for problem 15 is wrong. We suggest altering the problem as indicated in the errata.
Exercises 17--25 illustrate changes of variables to help in "linearization" near equilibrium points.
In Exercise 26, changing the independent variable is discussed.
Comments on the Labs
Technology for sketching solutions can and (we think) should be used for doing the labs, but students are sometimes reluctant.
Lab 1.1: 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 which many have not seen before. This lab can be started immediately after Section 1.1 is covered.
Lab 1.2: Logistic Population Models with Harvesting
The equations in this lab are considerably more complicated. 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.
Lab 1.3: Rate of Memorization Model
Because the model equation can be easily solved, 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 1.4: Euler's Method for Finding Roots of Polynomials
This lab can be started after Section 1.4, but it also works well with phase lines (Section 1.6).
Lab 1.5: 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 isn't 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.
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Version 1.1. May, 1996.