Differential Equations and Linear Algebra
Preface  PURPOSE AND PREREQUISITESThis book is intended as a textbook for a course in differential equations with linear algebra, to follow the differential and integral calculus. Since the syllabus of such a course is by no means standard, we have included more material than can be covered in a single course—possibly enough material for a two-semester course. This additional material is included to broaden the menu for the instructor and to increase the text's subsequent usefulness as a reference book for the student.Written for engineering, science, and computer science students, the approach is aimed at the applications oriented student but is also intended to be rigorous and to reveal the beauty and elegance of the subject.Why blend linear algebra with the differential equations? Since mid-twentieth century, the traditional course in differential equations has been offered in the first or second semester of the sophomore year and has relied on only a minimum of linear algebra, most notably the use of determinants. More recently, beginning with the advent of digital computers on campuses and in industry around the 1960s, a course or part of a course in linear algebra has become a part of most engineering science curricula. Given the current interest in introducing linear algebra earlier in curricula, the growing importance of systems of differential equations, and the natural use of linear algebra concepts in the study of differential equations, it seems best to move toward an integrated approach.   FLEXIBILITYThe text is organized so as to be flexible. For instance, it is generally considered desirable to include some nonlinear phase plane analysis in a course on differential equations since the qualitative topological approach complements the traditional analytical approach and also powerfully emphasizes the differences between linear and nonlinear systems. However, that topic usually proves to be a "luxury" to which one can devote one or two classes at best. Thus, we have arranged the phase plane material to allow anywhere from a one-class introduction to a moderately detailed discussion: We introduce the phase plane in only four pages in Section 7.3 in support of our discussion of the harmonic oscillator and we return to it in Chapter 11. There, Section 11.2 affords a more detailed overview of the method and provides another possible stopping point.To assist the instructor in the syllabus design we list some sections and subsections as optional but emphasize that these designations are subjective and intended only as a rule of thumb. (To the student we note that "optional" is not intended to mean unimportant, but only as a guide as to which material can be omitted by virtue of not being a prerequisite for the material that follows.)   SPECIFIC PEDAGOGICAL DECISIONSSeveral pedagogical decisions made in writing this text deserve explanation.Chapter sequence: Some instructors prefer to discuss numerical solution early, even within the study of first-order equations. Placement of the material on numerical solution near the end of this text does not rule out such an approach for one could cover Sections 12.1-12.2 on Eider's method, say, at any point in Chapters 2 or 3. Here, it seemed preferable to group Chapters 11 (on the phase plane) and 12 (on numerical solution) together since they complement the analytical approach, the former being qualitative and the latter being quantitative. As such, these two chapters might well have been made the final chapters, with the Laplace transform chapter moving up to precede or to follow Chapter 8 on series solution. Such movement is possible in a course syllabus since other chapters do not depend on series solution or on the Laplace transform. Also along these lines, it might seem awkward that Chapters 4 and 5 on vectors and matrices are separated from Chapter 9 on the eigenvalue problem. This separation may not be as great as it appears since in a one-semester course Chapter 8 might well be omitted. In any case in a combined approach to differential equations and linear algebra it seems logical to intersperse these two topics as naturally as possible rather than presenting them end-to-end. It may even be true that for optimal student retention it is good to have a gap between first meeting the linear algebra in Chapters 4 and 5 and returning to it in Chapter 9, so that it feels more like one is studying the subject twice.  Placing Gauss elimination in Chapter 4 on vectors rather than in Chapter 5 on matrices and linear algebraic equations: Just as one studies the real number axis before studying functions (mappings from one such axis to another), it seems appropriate to study vector spaces before studying matrices (which provide mappings from one vector space to another). In that case we find—in discussing span, linear dependence, bases, and expansions in Chapter 4—that we need to solve systems of coupled linear algebraic equations. Hence, we devote Section 4.5, which precedes that discussion, to Gauss elimination.  Introducing the Heaviside function in the chapter on first-order differential equations rather than in the chapter on the Laplace transform: If the forcing function is given piecewise, solution of the differential equation by a computer algebra system (Maple in this text) requires us to give a single expression for that function, and that can be accomplished using the Heaviside function. Further, including the Heaviside function in Chapter 2 makes it possible to include that topic even if the chapter on the Laplace transform is not covered.Computer Algebra SystemAs a representative computer algebra system this text uses Maple, but does not assume prior knowledge of that system. The Maple discussion is confined to subsections at the end of most sections, immediately preceding the exercises; see, for example, Sections 2.2 and 2.3. The reader can bypass those discussions entirely since they are supplemental and intended to show the student how to carry out various Maple calculations relevant to the material in that section. In some cases they explain how text figures were generated. The view represented here is that it would be foolish not to use the powerful computer algebra systems that are now available, but that primary emphasis should continue to rest firmly on fundamentals and understanding of the theory and methods. See also the section on supplements, below.EXERCISESEnd-of-section exercises are of different kinds and are arranged, typically, as follows. First, and usually near the beginning of the exercise group, are exercises that follow up on small gaps in the reading, thus engaging the student more fully in the reading (e.g., Exercises 1 and 2 of Section 3.5). Second, there are usually numerous "drill" type exercises that ask the student to mimic steps or calculations that are essentially similar to those demonstrated in the text (e.g., there are 19 matrices to invert by hand in Exercise 1 of Section 5.6). Third, there are exercises that call for the use of Maple (e.g., Exercise 3 of Section 5.6 and Exercise 4 of Section 10.4). Fourth, some exercises involve physical applications (e.g., Exercise 22 of Section 2.4 on the distribution of a pollutant in a river, Exercises 17 and 18 of Section 5.6 on electrical circuits, and Exercise 14 of Section 5.8 on computer graphics). And, fifth, there are exercises intended to extend the text and increase its value as a reference book (e.g., Exercises 7-12 of Section 2.3 on the Bernoulli, Riccati, Alembert-Lagrange, and Clairaut equations, and Exercise 2 of Section 3.3 on envelopes). Answers to selected exercises (which are denoted in the text by underlining the equation number) are given at the end of the book.SYLLABUS DESIGNDesigning a two-semester course is simple in the sense that one would probably cover virtually everything in the text. Thus, let us restrict our comments to the design of a one-semester course. As a general comment we note that sections and subsections are arranged with an eye toward flexibility. In Chapter 10, for instance, one could limit the coverage to Sections 10.1—10.3 or one could cover Sections 10.1, 10.2, and 10.4. As a specific example, at the University of Delaware mechanical engineers are currently required to take a three-course sequence in their sophomore year as follows. In the fall they take a three-credit course on differential equations and linear algebra following a syllabus somewhat as follows: Chapters 1-7 and 9-10 with these sections omitted-2.3.3, 2.4.2, 3.4, 4.4.2, 4.4.3, 4.5.6, 4.5.7, 4.8.3, 4.9.4, 4.9.5, 5.6.5, 5.7.2, 5.8, 6.6.2, 6.7.3, 6.7.4, 9.4.2, 9.4.3, 10.3.3, and 10.5-10.7.In the Spring they take two more courses, one covering Laplace transforms, field theory, and partial differential equations, and the other covering numerical methods, including the numerical solution of ordinary and partial differential equations.SUPPLEMENTSFor information regarding the Instructor's Solution Manual and other supplements, see the publisher's website, available 1/1/O1 at prenhall/greenber