Differential Equations With Boundary-value Problems With Ilrn Tutorial

by Zill, Dennis G.; Cullen, Michael R.

• ISBN13: 9780534418878
• ISBN10: 0534418872
• Edition: 6th
• Format: Paperback
• Copyright: 2004-10-19
• Publisher: Brooks Cole

Summary

Now enhanced with the innovative DE Tools CD-ROM and the iLrn teaching and learning system, this proven text explains the "how" behind the material and strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This accessible text speaks to students through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and group projects. This book was written with the student's understanding firmly in mind. Using a straightforward, readable, and helpful style, this book provides a thorough treatment of boundary-value problems and partial differential equations.

Table of Contents

Preface	p. xi
Acknowledgments	p. xv
Introduction to Differential Equations	p. 1
Definitions and Terminology	p. 2
Initial-Value Problems	p. 15
Differential Equations as Mathematical Models	p. 22
Chapter 1 in Review	p. 37
First-Order Differential Equations	p. 39
Solution Curves Without the Solution	p. 40
Separable Variables	p. 51
Linear Equations	p. 60
Exact Equations	p. 72
Solutions by Substitutions	p. 80
A Numerical Solution	p. 86
Chapter 2 in Review	p. 92
Modeling with First-Order Differential Equations	p. 95
Linear Equations	p. 96
Nonlinear Equations	p. 109
Systems of Linear and Nonlinear Differential Equations	p. 121
Chapter 3 in Review	p. 130
Project Module: Harvesting of Renewable Natural Resources	p. 133
Higher-Order Differential Equations	p. 138
Preliminary Theory: Linear Equations	p. 139
Initial-Value and Boundary-Value Problems	p. 139
Homogeneous Equations	p. 142
Nonhomogeneous Equations	p. 148
Reduction of Order	p. 154
Homogeneous Linear Equations with Constant Coefficients	p. 158
Undetermined Coefficients--Superposition Approach	p. 167
Undetermined Coefficients--Annihilator Approach	p. 178
Variation of Parameters	p. 188
Cauchy-Euler Equation	p. 193
Solving Systems of Linear Equations by Elimination	p. 201
Nonlinear Equations	p. 207
Chapter 4 in Review	p. 212
Modeling with Higher-Order Differential Equations	p. 215
Linear Equations: Initial-Value Problems	p. 216
Spring/Mass Systems: Free Undamped Motion	p. 216
Spring/Mass Systems: Free Damped Motion	p. 220
Spring/Mass Systems: Driven Motion	p. 224
Series Circuit Analogue	p. 227
Linear Equations: Boundary-Value Problems	p. 237
Nonlinear Equations	p. 247
Chapter 5 in Review	p. 259
Project Module: The Collapse of the Tacoma Narrows Suspension Bridge	p. 263
Series Solutions of Linear Equations	p. 267
Solutions About Ordinary Points	p. 268
Review of Power Series	p. 268
Power Series Solutions	p. 271
Solutions About Singular Points	p. 280
Two Special Equations	p. 292
Chapter 6 in Review	p. 304
The Laplace Transform	p. 306
Definition of the Laplace Transform	p. 307
Inverse Transform and Transforms of Derivatives	p. 314
Translation Theorems	p. 324
Translation on the s-Axis	p. 324
Translation on the t-Axis	p. 328
Additional Operational Properties	p. 338
Dirac Delta Function	p. 351
Systems of Linear Equations	p. 354
Chapter 7 in Review	p. 361
Systems of Linear First-Order Differential Equations	p. 364
Preliminary Theory	p. 365
Homogeneous Linear Systems with Constant Coefficients	p. 375
Distinct Real Eigenvalues	p. 376
Repeated Eigenvalues	p. 380
Complex Eigenvalues	p. 384
Variation of Parameters	p. 393
Matrix Exponential	p. 399
Chapter 8 in Review	p. 404
Project Module: Earthquake Shaking of Multistory Buildings	p. 406
Numerical Solutions of Ordinary Differential Equations	p. 410
Euler Methods and Error Analysis	p. 411
Runge-Kutta Methods	p. 417
Multistep Methods	p. 424
Higher-Order Equations and Systems	p. 427
Second-Order Boundary-Value Problems	p. 433
Chapter 9 in Review	p. 438
Plane Autonomous Systems and Stability	p. 439
Autonomous Systems, Critical Points, and Periodic Solutions	p. 440
Stability of Linear Systems	p. 448
Linearization and Local Stability	p. 458
Modeling Using Autonomous Systems	p. 470
Chapter 10 in Review	p. 480
Orthogonal Functions and Fourier Series	p. 483
Orthogonal Functions	p. 484
Fourier Series	p. 489
Fourier Cosine and Sine Series	p. 495
Sturm-Liouville Problem	p. 504
Bessel and Legendre Series	p. 511
Fourier-Bessel Series	p. 512
Fourier-Legendre Series	p. 515
Chapter 11 in Review	p. 519
Partial Differential Equations and Boundary-Value Problems in Rectangular Coordinates	p. 521
Separable Partial Differential Equations	p. 522
Classical Equations and Boundary-Value Problems	p. 527
Heat Equation	p. 533
Wave Equation	p. 536
Laplace's Equation	p. 542
Nonhomogeneous Equations and Boundary Conditions	p. 547
Orthogonal Series Expansions	p. 551
Boundary-Value Problems Involving Fourier Series in Two Variables	p. 555
Chapter 12 in Review	p. 559
Boundary-Value Problems in other Coordinate Systems	p. 561
Problems Involving Laplace's Equation in Polar Coordinates	p. 562
Problems in Polar and Cylindrical Coordinates: Bessel Functions	p. 567
Problems in Spherical Coordinates: Legendre Polynomials	p. 575
Chapter 13 in Review	p. 578
Integral Transform Method	p. 581
Error Function	p. 582
Applications of the Laplace Transform	p. 584
Fourier Integral	p. 595
Fourier Transforms	p. 601
Chapter 14 in Review	p. 607
Numerical Solutions of Partial Differential Equations	p. 610
Elliptic Equations	p. 611
Parabolic Equations	p. 617
Hyperbolic Equations	p. 625
Chapter 15 in Review	p. 630
Appendixes	p. 1
Gamma Function	p. 1
Introduction to Matrices	p. 3
Laplace Transforms	p. 25
Selected Answers for Odd-Numbered Problems	p. 1
Index	p. 1
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