Elementary Differential Equations with Boundary Value Problems

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

• ISBN13: 9780131457744
• ISBN10: 0131457748
• Edition: 5th
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: Pearson College Div

Summary

This accessible, attractive, and interesting book enables readers to first solve those differential equations that have the most frequent and interesting applications. This approach illustrates the standard elementary techniques of solution of differential equations. Precise and clear-cut statements of fundamental existence and uniqueness theorems allow understanding of their role in this subject. A strong numerical approach emphasizes that the effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques.The first few sections of most chapters introduce the principle ideas of each topic, with remaining sections devoted to extensions and applications. Topics covered include first-order differential equations, linear equations of higher order, power series methods, Laplace transform methods, linear systems of differential equations, numerical methods, nonlinear systems and phenomena, Fourier series methods, and Eigenvalues and boundary value problems.For those involved in the fields of science, engineering, and mathematics.

Table of Contents

Preface	vii
Chapter 1 First-Order Differential Equations	1	(95)
1.1 Differential Equations and Mathematical Models	1	(9)
1.2 Integrals as General and Particular Solutions	10	(8)
1.3 Slope Fields and Solution Curves	18	(13)
1.4 Separable Equations and Applications	31	(14)
1.5 Linear First-Order Equations	45	(12)
1.6 Substitution Methods and Exact Equations	57	(14)
1.7 Population Models	71	(11)
1.8 Acceleration-Velocity Models	82	(14)
Chapter 2 Linear Equations of Higher Order	96	(92)
2.1 Introduction: Second-Order Linear Equations	96	(13)
2.2 General Solutions of Linear Equations	109	(11)
2.3 Homogeneous Equations with Constant Coefficients	120	(11)
2.4 Mechanical Vibrations	131	(13)
2.5 Nonhomogeneous Equations and Undetermined Coefficients	144	(13)
2.6 Forced Oscillations and Resonance	157	(11)
2.7 Electrical Circuits	168	(8)
2.8 Endpoint Problems and Eígenvalues	176	(12)
Chapter 3 Power Series Methods	188	(71)
3.1 Introduction and Review of Power Series	188	(13)
3.2 Series Solutions Near Ordinary Points	201	(11)
3.3 Regular Singular Points	212	(15)
3.4 Method of Frobenius: The Exceptional Cases	227	(14)
3.5 Bessel's Equation	241	(9)
3.6 Applications of Bessel Functions	250	(9)
Chapter 4 Laplace Transform Methods	259	(58)
4.1 Laplace Transforms and Inverse Transforms	259	(10)
4.2 Transformation of Initial Value Problems	269	(11)
4.3 Translation and Partial Fractions	280	(7)
4.4 Derivatives, Integrals, and Products of Transforms	287	(8)
4.5 Periodic and Piecewise Continuous Input Functions	295	(11)
4.6 Impulses and Delta Functions	306	(11)
Chapter 5 Linear Systems of Differential Equations	317	(103)
5.1 First-Order Systems and Applications	317	(12)
5.2 The Method of Elimination	329	(9)
5.3 Matrices and Linear Systems	338	(19)
5.4 The Eigenvalue Method for Homogeneous Systems	357	(15)
5.5 Second-Order Systems and Mechanical Applications	372	(11)
5.6 Multiple Eigenvalue Solutions	383	(16)
5.7 Matrix Exponentials and Linear Systems	399	(12)
5.8 Nonhomogeneous Linear Systems	411	(9)
Chapter 6 Numerical Methods	420	(50)
6.1 Numerical Approximation: Euler's Method	420	(12)
6.2 A Closer Look at the Euler Method	432	(11)
6.3 The Runge-Kutta Method	443	(10)
6.4 Numerical Methods for Systems	453	(17)
Chapter 7 Nonlinear Systems and Phenomena	470	(74)
7.1 Equilibrium Solutions and Stability	470	(8)
7.2 Stability and the Phase Plane	478	(12)
7.3 Linear and Almost Linear Systems	490	(13)
7.4 Ecological Models: Predators and Competitors	503	(13)
7.5 Nonlinear Mechanical Systems	516	(16)
7.6 Chaos in Dynamical Systems	532	(12)
Chapter 8 Fourier Series Methods	544	(67)
8.1 Periodic Functions and Trigonometric Series	544	(9)
8.2 General Fourier Series and Convergence	553	(7)
8.3 Fourier Sine and Cosine Series	560	(10)
8.4 Applications of Fourier Series	570	(5)
8.5 Heat Conduction and Separation of Variables	575	(13)
8.6 Vibrating Strings and the One-Dimensional Wave Equation	588	(12)
8.7 Steady-State Temperature and Laplace's Equation	600	(11)
Chapter 9 Eigenvalues and Boundary Value Problems	611	(61)
9.1 Sturm-Liouville Problems and Eigenfunction Expansions	611	(11)
9.2 Applications of Eigenfunction Series	622	(9)
9.3 Steady Periodic Solutions and Natural Frequencies	631	(8)
9.4 Cylindrical Coordinate Problems	639	(14)
9.5 Higher-Dimensional Phenomena	653	(19)
References for Further Study	672	(3)
Appendix: Existence and Uniqueness of Solutions	675	(14)
Answers to Selected Problems	689
Index	I-1

Excerpts

We wrote this book to provide a concrete and readable text for the traditional course in elementary differential equations that science, engineering, and mathematics students take following calculus. Our treatment is shaped throughout by the goal of a flexible exposition that students will find accessible, attractive, and interesting. The applications of differential equations have played a singular role in the historical development of the subject, and whole areas of study exist mainly because of their applications. We therefore want our students to learn first to solve those differential equations that enjoy the most frequent and interesting applications. Hence we make consistent use of appealing applications to motivate and illustrate the standard elementary techniques of solution of differential equations. 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. The list of introductory topics in differential equations is quite standard, and a glance at our chapter titles will reveal no major surprises, though in the fine structure we have attempted to add a bit of zest here and there. In most chapters the principal ideas of the topic are introduced in the first few sections of the chapter, and the remaining sections are devoted to extensions and applications. Hence the instructor has a wide range of choice regarding breadth and depth of coverage. At various points our approach reflects the widespread use of computer programs for the numerical solution of differential equations. Nevertheless, we continue to believe that the traditional elementary analytical methods of solution are important for students to learn. One reason is that 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. Fifth Edition Features This text is an extensive revision ofElementary Differential Equations with Boundary Value Problems,Fourth Edition. Among the new and enhanced features: Almost 20% of the text's over 1900problemsare new for this edition or are newly revised to include graphic or qualitative content. Additional text and discussion, with about 15% of the workedexamplesin the book being new or newly revised for this edition. Almost 700computer-generated figures--over half of them new for this edition and most constructed using Mathematica or MATLAB--show students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life. For instance, the cover graphic is a color enhanced version of Fig. 4.3.5 in the text, illustrating the damped forced oscillations of a certain mass-spring system. About 15 new or revisedapplication modulesfollow appropriate sections in the book. Their purpose is to add concrete applied emphasis and to engage students in more extensive investigations than afforded by typical exercises and problems. Asolid numerical emphasisis provided in Chapter 6 on Numerical Methods by the inclusion of generic numerical algorithms and a limited number of illustrative graphing calculator, BASIC, and MATLAB routines. A contemporary perspective--shaped by the availability of computational aids--permits a more streamlined (though still complete) coverage of certain standard topics (like ex