Elementary Differential Equations with Boundary Value Problems

Description

New and updated features of this title

• Chapter 1: New Figures 1.3.9 and 1.3.10 showing direction fields that illustrate failure of existence and uniqueness of solutions New Problems 34 and 35 showing that small changes in initial conditions can make big differences in results, but big changes in initial conditions may sometimes make only small differences in results; new Remarks 1 and 2 clarifying the concept of implicit solutions; new Remark clarifying the meaning of homogeneity for first-order equations; additional details inserted in the derivation of the rocket propulsion equationand new Problem 5 inserted to investigate the liftoff pause of a rocket on the launch pad sometimes observed before blastoff.
• Chapter 2: New explanation of signs and directions of internal forces in mass-spring systems; new introduction of differential operators and clarification of the algebra of polynomial operators; new introduction and illustration of polar exponential forms of complex numbers; fuller explanation of method of undetermined coefficients in Examples 1 and 3; new Remarks 1 and 2 introducing “shooting” terminology, and new Figures 2.8.1 and 2.8.2 illustrating why some endpoint value problems have infinitely many solutions, while others have no solutions at all; new Figures 2.8.4 and 2.8.5 illustrating different types of eigenfunctions.
• Chapter 3: New Problem 35 on determination of radii of convergence of power series solutions of differential equations; new Example 3 just before the subsection on logarithmic cases in the method of Frobenius, to illustrate first the reduction-of-order formula with a simple non-series problem. Chapter 4: New discussion clarifying functions of exponential order and existence of Laplace transforms; new Remark discussing the mechanics of partial-fraction decomposition; new much-expanded discussion of the proof of the Laplace-transform existence theorem and its extension to include the jump discontinuities that play an important role in many practical applications.
• Chapter 5: New Problems 20-23 for student exploration of three-railway-cars systems with different initial velocity conditions; new Remark illustrating the relation between matrix exponential methods and the generalized eigenvalue methods discussed previously; new exposition inserted at end of section to explain the connection between matrix variation of parameters here and (scalar) variation of parameters for second-order equations discussed previously in Chapter 3.
• Chapter 6: New discussion with new Figures 6.3.11 and 6.3.12 clarifying the difference between rotating and non-rotating coordinate systems in moon-earth orbit problems. Chapter 7: New remarks on phase plane portraits, autonomous systems, and critical points; new introduction of linearized systems; new 3-dimensional Figures 6.5.18 and 6.5.20 illustrating Lorenz and Rössler trajectories.
• Chapter 8: New considerably expanded explanation of even and odd extensions and their Fourier sine-cosine series; new discussion of periodic and non-periodic particular solutions illustrated by new Figure 8.4.4, together with new Problems 19 and 20 at end of section; new example discussion inserted at end of section to illustrate the effects of damping in mass-spring systems; new discussion of signs and direction of heat flow in the derivation of the heat equation. Chapter 9: Clarification of the effect of internal stretching in deriving the wave equation for longitudinal vibrations of a bar; new Figures 9.5.15 and 9.5.16 illustrating ocean waves on a small planet.

Table of Contents

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia’s honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution’s highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with- Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran’s Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee’s research team’s primary focus was on the active transport of sodium ions by biological membranes. Penney’s primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

Hallmark features of this title

• Proven chapter and section structure of the book is unchanged: Instructors’ notes and syllabi will not require revision to continue teaching with the Sixth Edition.
• A solid numerical emphasis includes generic numerical algorithms and a limited number of illustrative graphic calculator, BASIC, and MATLAB routines.
• Wide range of choices regarding breadth and depth of coverage: The first few sections of most chapters introduce the principal ideas of each topic, with remaining sections devoted to extensions and applications.
• Computer-generated figures include graphics using Mathematica or MATLAB, showing students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life.
• Application Modules follow appropriate sections in the book, giving students concrete applied emphasis and engages them in more extensive investigations than in typical exercises and problems.

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