Differential Equations and Boundary Value Problems : Computing and Modeling

by Edwards, C. Henry; Penney, David E.

• ISBN13: 9780130652454
• ISBN10: 0130652458
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: Pearson College Div

Summary

This practical book reflects the new technological emphasis that permeates differential equations, including the wide availability of scientific computing environments likeMaple, Mathematica, and MATLAB; it does not concentrate on traditional manual methods but rather on new computer-based methods that lead to a wider range of more realistic applications. The book starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the book. For mathematicians and those in the field of computer science.

Table of Contents

Application Modules	viii
Preface	xi
First-Order Differential Equations	1	(76)
Differential Equations and Mathematical Models	1	(9)
Integrals as General and Particular Solutions	10	(8)
Slope Fields and Solution Curves	18	(13)
Separable Equations and Applications	31	(15)
Linear First-Order Equations	46	(12)
Substitution Methods and Exact Equations	58	(19)
Mathematical Models and Numerical Methods	77	(67)
Population Models	77	(13)
Equilibrium Solutions and Stability	90	(8)
Acceleration-Velocity Models	98	(12)
Numerical Approximation: Euler's Method	110	(12)
A Closer Look at the Euler Method	122	(10)
The Runge-Kutta Method	132	(12)
Linear Equations of Higher Order	144	(98)
Introduction: Second-Order Linear Equations	144	(14)
General Solutions of Linear Equations	158	(12)
Homogeneous Equations with Constant Coefficients	170	(12)
Mechanical Vibrations	182	(13)
Nonhomogeneous Equations and Undetermined Coefficients	195	(14)
Forced Oscillations and Resonance	209	(13)
Electrical Circuits	222	(7)
Endpoint Problems and Eigenvalues	229	(13)
Introduction to Systems of Differential Equations	242	(39)
First-Order Systems and Applications	242	(12)
The Method of Elimination	254	(11)
Numerical Methods for Systems	265	(16)
Linear Systems of Differential Equations	281	(85)
Matrices and Linear Systems	281	(19)
The Eigenvalue Method for Homogeneous Systems	300	(15)
Second-Order Systems and Mechanical Applications	315	(13)
Multiple Eigenvalue Solutions	328	(16)
Matrix Exponentials and Linear Systems	344	(14)
Nonhomogeneous Linear Systems	358	(8)
Nonlinear Systems and Phenomena	366	(69)
Stability and the Phase Plane	366	(12)
Linear and Almost Linear Systems	378	(15)
Ecological Models: Predators and Competitors	393	(13)
Nonlinear Mechanical Systems	406	(17)
Chaos in Dynamical Systems	423	(12)
Laplace Transform Methods	435	(62)
Laplace Transforms and Inverse Transforms	435	(11)
Transformation of Initial Value Problems	446	(11)
Translation and Partial Fractions	457	(10)
Derivatives, Integrals, and Products of Transforms	467	(8)
Periodic and Piecewise Continuous Input Functions	475	(11)
Impulses and Delta Functions	486	(11)
Power Series Methods	497	(75)
Introduction and Review of Power Series	497	(13)
Series Solutions Near Ordinary Points	510	(13)
Regular Singular Points	523	(16)
Method of Frobenius: The Exceptional Cases	539	(15)
Bessel's Equation	554	(9)
Applications of Bessel Functions	563	(9)
Fourier Series Methods	572	(73)
Periodic Functions and Trigonometric Series	572	(9)
General Fourier Series and Convergence	581	(8)
Fourier Sine and Cosine Series	589	(12)
Applications of Fourier Series	601	(5)
Heat Conduction and Separation of Variables	606	(15)
Vibrating Strings and the One-Dimensional Wave Equation	621	(14)
Steady-State Temperature and Laplace's Equation	635	(10)
Eigenvalues and Boundary Value Problems	645	(66)
Sturm-Liouville Problems and Eigenfunction Expansions	645	(13)
Applications of Eigenfunction Series	658	(10)
Steady Periodic Solutions and Natural Frequencies	668	(10)
Cylindrical Coordinate Problems	678	(15)
Higher-Dimensional Phenomena	693	(18)
References for Further Study	711	(3)
Appendix: Existence and Uniqueness of Solutions	714	(15)
Answers to Selected Problems	729
Index	1

Excerpts

Many introductory differential equations courses in the recent past have emphasized the formal solution of standard types of differential equations using a (seeming) grab-bag of systematic solution techniques. Many students have concentrated on learning to match memorized methods with memorized equations. The evolution of the present text is based on experience teaching a course with a greater emphasis on conceptual ideas and the use of applications and computing projects to involve students in more intense and sustained problem-solving experiences. The availability of technical computing environments likeMaple, Mathematica,and MATLAB is reshaping the role and applications of differential equations in science and engineering and has shaped our approach in this text. New technology motivates a shift in emphasis from traditional manual methods to both qualitative and computer-based methods that render accessible a wider range of more realistic applications; permit the use of both numerical computation and graphical visualization to develop greater conceptual understanding; and encourage empirical investigations that involve deeper thought and analysis than standard textbook problems. Major Features The following features of this text are intended to support a contemporary differential equations course that augments traditional core skills with conceptual perspectives that students will need for the effective use of differential equations in their subsequent work and study: Coverage of seldom-used topics has been trimmed and new topics added to place a greater emphasis on core techniques as well as qualitative aspects of the subject associated with direction fields, solution curves, phase plane portraits, and dynamical systems. We combine symbolic, graphic, and numeric solution methods wherever it seems advantageous. A fresh computational flavor should be evident in figures, examples, problems, and applications throughout the text. About 15% of the examples in the text are new or newly revised for this edition. The organization of the book places an increased emphasis on linear systems of differential equations, which are covered in Chapters 4 and 5 (together with the necessary linear algebra), followed by a substantial treatment in Chapter 6 of nonlinear systems and phenomena (including chaos in dynamical systems). This book begins and ends with discussions and examples of the mathematical modeling of real-world phenomena. Students learn through mathematical modeling and empirical investigation to balance the questions of what equation to formulate, how to solve it, and whether a solution will yield useful information.,/LI> The first course in differential equations should also be a window on the world of mathematics. While it is neither feasible nor desirable to include proofs of the fundamental existence and uniqueness theorems along the way in an elementary course, students need to see precise and clear-cut statements of these theorems and to understand their role in the subject. We include appropriate existence and uniqueness proofs in the Appendix and occasionally refer to them in the main body of the text. While our approach reflects the widespread use of new computer methods for the solution of differential equations, certain elementary analytical methods of solution (as in Chapters 1 and 3) are important for students to learn. Effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques; the construction of a realistic numerical model often is based on the study of a simpler analytical model. We therefore continue to stress the mastery of traditional solution techniques (especially through the inclusion of extensive problem sets). Computing Features The following features highlight the flavor of computing technology that distinguishes much of