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Elementary Differential Equations with Boundary Value Problems,9780131457744
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Elementary Differential Equations with Boundary Value Problems

by ;
Edition:
6th
ISBN13:

9780131457744

ISBN10:
0131457748
Format:
Hardcover
Pub. Date:
1/1/2008
Publisher(s):
Pearson College Div
List Price: $144.20
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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


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