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DvvOverview of Chapter One
Two of our major goals throughout the text are
to give a balanced approach to analytic, qualitative, and numerical techniques for studying solutions and
to make the applications of the subject as transparent as possible.
To accomplish these goals, the textbook continually introduces new models and modifies old ones using all three types of techniques whenever possible. Chapter 1 introduces, by example, all of these themes for first-order equations with emphasis placed 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. In any case, Section 1.6 (phase lines) is very important to the development of systems in Chapter 2.
1.1 Modeling Via Differential Equations
The 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, there is a connection between the assumptions made regarding the physical system and the individual terms in the differential equation. In studying solutions, the formula and/or the graph of a solution provides a prediction about the future behavior of the physical system under consideration. It also emphasizes 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 1-2 begin the process of thinking qualitatively about differential equations.
Exercises 5-7 deal with a simple learning model (see also Lab 1.1).
Exercises 13, 14 are "do-able" modeling problems. In general, students find modifying existing models easier than creating new ones from scratch.
Exercises 18-21 deal with systems of differential equations, 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
This section gives the first examples of the "analytic" approach (i.e., finding closed-form formulas for solutions) to differential equations. It begins 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. The only equations that considered at this point are separable, and 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. The students may need some review of their techniques of integration; the only real difficulty is in dealing with absolute value signs.
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
Wherever possible, a description of the long-term behavior of a solution is required to promote qualitative thinking.
In Exercise 3, students are asked to construct a differential equation given a solution. This is a change in point of view that may look difficult.
Exercises 10-18, 30-32 are standard examples of separable differential equations, and all of the integrals have closed-form expressions. 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.
Exercise 35 is fairly standard "word problems."
In Exercise 41, the first step is to determine the proportionality constant for Newton's law of cooling.
1.3 Qualitative Technique: Slope Fields
This section introduces the first geometric and qualitative technique-slope fields. It 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. It also encourages 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. RC circuits are introduced as an example in this section without developing the physics or the circuit theory.
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 1,5 are very easy with technology, but it is worthwhile to do them to gain intuition about what slope fields look like and to practice sketching solutions.
Exercises 7,9 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 are required to write a paragraph that describes what features of the slope fields they used to determine which is which.
Exercises 11, 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.
Exercise 21 involves the RC circuit model. They serve as a good advertisement for qualitative and numeric methods.
1.4 Numerical Technique: Euler's Method
This section introduces 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.
A careful analysis of error in Euler's method is given in Section 7.1, and improved Euler's method and Runge-Kutta are also presented in Chapter 7.
DETools: EulersMethod is a tool that illustrates Euler's method for a variety of equations. It is 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. 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?)
1.5 Existence and Uniqueness of Solutions
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.
The issue of the domain of definition of a solution-a theoretical point that has been ignored in Sections 1.1-1.4 is considered. Some students will have wondered about domains of definition in Section 1.2 where restricted domains are the norm. Here, emphasize is on 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 2,3,5-6 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 12-17 are exercises that address the domain of definition, and 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. This is not only necessary but good as well--the more ways of representing a differential equation we have, the more ways we have of obtaining information about its solutions.
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.
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.
Exercise 43 concerns the behavior of solutions near equilbria in cases where linearization is not conclusive.
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 the 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 familiarity 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. Exercise 12 illustrates the power of the ideas introduced in this section.
DETools: This section is the appropriate place to turn on the bifurcation plane in PhaseLines.
Comments on selected exercises
Exercises 3,5 consider one-parameter families to find the bifurcation values.
Exercises 9,13 return to population and harvesting examples.
Exercise 12 requires interpretation of bifurcation diagram. The conclusions are quite striking.
Exercise 15 uses the bifurcation plane feature in PhaseLines.
Exercises 22,23 involve some unusual bifurcations.
1.8 Linear Differential Equations
Linear (nonautonomous) equations occur in sufficiently many of the standard examples that it seems appropriate to introduce the technique of integrating factors. This section give motivation ideas behind integrating factors as much as possible.
Comments on selected exercises
Exercises 4-8,10-12 are standard. In some cases, the integration is challenging.
Exercises 15,18,20 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 Exercise 27 is mixing problem.
Exercise 29 considers the extreme case when the initial volume is zero (division by zero is a danger).
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