Differential Equations with Boundary Value Problems

by Polking, John; Boggess, Al; Arnold, David

• ISBN13: 9780131862364
• ISBN10: 0131862367
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-07-28
• Publisher: Pearson

Summary

Combining traditional material with a modern systems approach, this handbook provides a thorough introduction to differential equations, tempering its classic "pure math" approach with more practical applied aspects.Features up-to-date coverage of key topics such as first order equations, matrix algebra, systems, and phase plane portraits. Illustrates complex concepts through extensive detailed figures. Focuses on interpreting and solving problems through optional technology projects.For anyone interested in learning more about differential equations.

Table of Contents

Preface	p. ix
Introduction to Differential Equations	p. 1
Differential Equation Models	p. 2
The Derivative	p. 7
Integration	p. 10
First-Order Equations	p. 18
Differential Equations and Solutions	p. 18
Solutions to Separable Equations	p. 31
Models of Motion	p. 44
Linear Equations	p. 54
Mixing Problems	p. 64
Exact Differential Equations	p. 73
Existence and Uniqueness of Solutions	p. 90
Dependence of Solutions on Initial Conditions	p. 102
Autonomous Equations and Stability	p. 107
The Daredevil Skydiver	p. 120
Modeling and Applications	p. 123
Modeling Population Growth	p. 124
Models and the Real World	p. 138
Personal Finance	p. 143
Electrical Circuits	p. 152
The Spruce Budworm	p. 158
Social Security, Now or Later	p. 161
Second-Order Equations	p. 163
Definitions and Examples	p. 163
Second-Order Equations and Systems	p. 174
Linear, Homogeneous Equations with Constant Coefficients	p. 179
Harmonic Motion	p. 190
Inhomogeneous Equations; the Method of Undetermined Coefficients	p. 199
Variation of Parameters	p. 209
Forced Harmonic Motion	p. 215
Nonlinear Oscillators	p. 228
The Laplace Transform	p. 231
The Definition of the Laplace Transform	p. 232
Basic Properties of the Laplace Transform	p. 241
The Inverse Laplace Transform	p. 248
Using the Laplace Transform to Solve Differential Equations	p. 256
Discontinuous Forcing Terms	p. 266
The Delta Function	p. 280
Convolutions	p. 287
Summary	p. 298
Forced Harmonic Oscillators	p. 299
Numerical Methods	p. 301
Euler's Method	p. 302
Runge-Kutta Methods	p. 314
Numerical Error Comparisons	p. 322
Practical Use of Solvers	p. 327
A Cautionary Tale	p. 332
Numerical Error Comparison	p. 334
Matrix Algebra	p. 335
Vectors and Matrices	p. 335
The Geometry of Systems of Linear Equations	p. 348
Solving Systems of Equations	p. 353
Properties of Solution Sets	p. 362
Subspaces	p. 371
Determinants	p. 384
An Introduction to Systems	p. 396
Definitions and Examples	p. 396
Geometric Interpretation of Solutions	p. 405
Qualitative Analysis	p. 417
Linear Systems	p. 425
Long-Term Behavior of Solutions	p. 441
Linear Systems with Constant Coefficients	p. 444
Overview of the Technique	p. 444
Planar Systems	p. 452
Phase Plane Portraits	p. 466
Higher Dimensional Systems	p. 484
The Exponential of a Matrix	p. 492
Qualitative Analysis of Linear Systems	p. 510
Higher-Order Linear Equations	p. 515
Inhomogeneous Linear Systems	p. 528
Phase Plane Portraits	p. 538
Oscillations of Linear Molecules	p. 539
Nonlinear Systems	p. 545
The Linearization of a Nonlinear System	p. 545
Long-Term Behavior of Solutions	p. 559
Invariant Sets and the Use of Nullclines	p. 566
Long-Term Behavior of Solutions to Planar Systems	p. 574
Conserved Quantities	p. 586
Nonlinear Mechanics	p. 592
The Method of Lyapunov	p. 609
Predator-Prey Systems	p. 619
Human Immune Response to Infectious Disease	p. 631
Analysis of Competing Species	p. 634
Series Solutions to Differential Equations	p. 637
Review of Power Series	p. 638
Series Solutions Near Ordinary Points	p. 650
Legendre's Equation	p. 662
Types of Singular Points--Euler's Equation	p. 668
Series Solutions Near Regular Singular Points	p. 677
Solutions in the Exceptional Cases	p. 690
Bessel's Equation and Bessel Functions	p. 701
Fourier Series	p. 712
Computation of Fourier Series	p. 713
Convergence of Fourier Series	p. 724
Fourier Cosine and Sine Series	p. 733
The Complex Form of a Fourier Series	p. 738
The Discrete Fourier Transform and the FFT	p. 741
Partial Differential Equations	p. 750
Derivation of the Heat Equation	p. 750
Separation of Variables for the Heat Equation	p. 756
The Wave Equation	p. 768
Laplace's Equation	p. 778
Laplace's Equation on a Disk	p. 785
Sturm Liouville Problems	p. 791
Orthogonality and Generalized Fourier Series	p. 801
Temperature in a Ball--Legendre Polynomials	p. 810
Time Dependent PDEs in Higher Dimension	p. 814
Domains with Circular Symmetry--Bessel Functions	p. 821
Answers to Odd-Numbered Problems	p. 1
Index	p. 1
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