Differential Equations and Boundary Value Problems: Computing and Modeling

✍ Scribed by Edwards, Charles Henry

• Publisher: Pearson
• Year: 2014
• Tongue: English
• Leaves: 795
• Category: Library

✦ Synopsis

For introductory courses in Differential Equations.This best-selling text by these well-known authors blends the traditional algebra problem solving skills with the conceptual development and geometric visualization of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Acknowledgments......Page 13
CONTENTS......Page 6
Application Modules......Page 9
Preface......Page 10
1.1 Differential Equations and Mathematical Models......Page 14
1.2 Integrals as General and Particular Solutions......Page 23
1.3 Slope Fields and Solution Curves......Page 30
1.4 Separable Equations and Applications......Page 43
1.5 Linear First-Order Equations......Page 58
1.6 Substitution Methods and Exact Equations......Page 70
2.1 Population Models......Page 88
2.2 Equilibrium Solutions and Stability......Page 99
2.3 Acceleration–Velocity Models......Page 106
2.4 Numerical Approximation: Euler’s Method......Page 117
2.5 A Closer Look at the Euler Method......Page 129
2.6 The Runge–Kutta Method......Page 139
3.1 Introduction: Second-Order Linear Equations......Page 149
3.2 General Solutions of Linear Equations......Page 162
3.3 Homogeneous Equations with Constant Coefficients......Page 175
3.4 Mechanical Vibrations......Page 185
3.5 Nonhomogeneous Equations and Undetermined Coefficients......Page 197
3.6 Forced Oscillations and Resonance......Page 210
3.7 Electrical Circuits......Page 222
3.8 Endpoint Problems and Eigenvalues......Page 229
4.1 First-Order Systems and Applications......Page 241
4.2 The Method of Elimination......Page 252
4.3 Numerical Methods for Systems......Page 262
5.1 Matrices and Linear Systems......Page 277
5.2 The Eigenvalue Method for Homogeneous Systems......Page 295
5.3 A Gallery of Solution Curves of Linear Systems......Page 309
5.4 Second-Order Systems and Mechanical Applications......Page 335
5.5 Multiple Eigenvalue Solutions......Page 348
5.6 Matrix Exponentials and Linear Systems......Page 362
5.7 Nonhomogeneous Linear Systems......Page 376
6.1 Stability and the Phase Plane......Page 385
6.2 Linear and Almost Linear Systems......Page 396
6.3 Ecological Models: Predators and Competitors......Page 408
6.4 Nonlinear Mechanical Systems......Page 421
6.5 Chaos in Dynamical Systems......Page 439
7.1 Laplace Transforms and Inverse Transforms......Page 450
7.2 Transformation of Initial Value Problems......Page 460
7.3 Translation and Partial Fractions......Page 471
7.4 Derivatives, Integrals, and Products of Transforms......Page 480
7.5 Periodic and Piecewise Continuous Input Functions......Page 487
7.6 Impulses and Delta Functions......Page 497
8.1 Introduction and Review of Power Series......Page 507
8.2 Series Solutions Near Ordinary Points......Page 519
8.3 Regular Singular Points......Page 530
8.4 Method of Frobenius: The Exceptional Cases......Page 546
8.5 Bessel’s Equation......Page 560
8.6 Applications of Bessel Functions......Page 569
9.1 Periodic Functions and Trigonometric Series......Page 577
9.2 General Fourier Series and Convergence......Page 585
9.3 Fourier Sine and Cosine Series......Page 593
9.4 Applications of Fourier Series......Page 605
9.5 Heat Conduction and Separation of Variables......Page 610
9.6 Vibrating Strings and the One-Dimensional Wave Equation......Page 624
9.7 Steady-State Temperature and Laplace’s Equation......Page 638
10.1 Sturm–Liouville Problems and Eigenfunction Expansions......Page 648
10.2 Applications of Eigenfunction Series......Page 660
10.3 Steady Periodic Solutions and Natural Frequencies......Page 670
10.4 Cylindrical Coordinate Problems......Page 679
10.5 Higher-Dimensional Phenomena......Page 694
References for Further Study......Page 711
Appendix: Existence and Uniqueness of Solutions......Page 714
Answers to Selected Problems......Page 728
C......Page 787
E......Page 788
I......Page 789
N......Page 790
R......Page 791
T......Page 792
Z......Page 793

✦ Subjects

Science;Mathematics;Textbooks;Computer Science;Computers;Engineering;Reference