Differential Equations Theory, Technique, and Practice

ISBN-10: 0072863153
ISBN-13: 9780072863154
Edition: 2007
List price: $157.50
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Book details

List price: $157.50
Copyright year: 2007
Publisher: McGraw-Hill Higher Education
Publication date: 1/4/2006
Binding: Hardcover
Pages: 544
Size: 6.50" wide x 9.25" long x 1.00" tall
Weight: 1.848
Language: English

Prefacep. VII
What is a Differential Equation?p. 1
Introductory Remarksp. 2
The Nature of Solutionsp. 4
Separable Equationsp. 10
First-Order Linear Equationsp. 13
Exact Equationsp. 17
Orthogonal Trajectories and Families of Curvesp. 22
Homogeneous Equationsp. 26
Integrating Factorsp. 29
Reduction of Orderp. 33
Dependent Variable Missingp. 33
Independent Variable Missingp. 35
The Hanging Chain and Pursuit Curvesp. 38
The Hanging Chainp. 38
Pursuit Curvesp. 42
Electrical Circuitsp. 45
Anatomy of an Application: The Design of a Dialysis Machinep. 49
Problems for Review and Discoveryp. 53
Second-Order Linear Equationsp. 57
Second-Order Linear Equations with Constant Coefficientsp. 58
The Method of Undetermined Coefficientsp. 63
The Method of Variation of Parametersp. 67
The Use of a Known Solution to Find Anotherp. 71
Vibrations and Oscillationsp. 75
Undamped Simple Harmonic Motionp. 75
Damped Vibrationsp. 77
Forced Vibrationsp. 80
A Few Remarks about Electricityp. 82
Newton's Law of Gravitation and Kepler's Lawsp. 84
Kepler's Second Lawp. 86
Kepler's First Lawp. 87
Kepler's Third Lawp. 89
Higher Order Linear Equations, Coupled Harmonic Oscillatorsp. 93
Historical Note: Eulerp. 99
Anatomy of an Application: Bessel Functions and the Vibrating Membranep. 101
Problems for Review and Discoveryp. 105
Qualitative Properties and Theoretical Aspectsp. 109
Review of Linear Algebrap. 110
Vector Spacesp. 110
The Concept of Linear Independencep. 111
Basesp. 113
Inner Product Spacesp. 114
Linear Transformations and Matricesp. 115
Eigenvalues and Eigenvectorsp. 117
A Bit of Theoryp. 119
Picard's Existence and Uniqueness Theoremp. 125
The Form of a Differential Equationp. 125
Picard's Iteration Techniquep. 126
Some Illustrative Examplesp. 127
Estimation of the Picard Iteratesp. 129
Oscillations and the Sturm Separation Theoremp. 130
The Sturm Comparison Theoremp. 138
Anatomy of an Application: The Green's Functionp. 142
Problems for Review and Discoveryp. 146
Power Series Solutions and Special Functionsp. 149
Introduction and Review of Power Seriesp. 150
Review of Power Seriesp. 150
Series Solutions of First-Order Differential Equationsp. 159
Second-Order Linear Equations: Ordinary Pointsp. 164
Regular Singular Pointsp. 171
More on Regular Singular Pointsp. 177
Gauss's Hypergeometric Equationp. 184
Historical Note: Gaussp. 189
Historical Note: Abelp. 190
Anatomy of an Application: Steady-State Temperature in a Ballp. 192
Problems for Review and Discoveryp. 194
Fourier Series: Basic Conceptsp. 197
Fourier Coefficientsp. 198
Some Remarks about Convergencep. 207
Even and Odd Functions: Cosine and Sine Seriesp. 211
Fourier Series on Arbitrary Intervalsp. 218
Orthogonal Functionsp. 221
Historical Note: Riemannp. 225
Anatomy of an Application: Introduction to the Fourier Transformp. 227
Problems for Review and Discoveryp. 236
Partial Differential Equations and Boundary Value Problemsp. 239
Introduction and Historical Remarksp. 240
Eigenvalues, Eigenfunctions, and the Vibrating Stringp. 243
Boundary Value Problemsp. 243
Derivation of the Wave Equationp. 244
Solution of the Wave Equationp. 246
The Heat Equationp. 251
The Dirichlet Problem for a Discp. 256
The Poisson Integralp. 259
Sturm-Liouville Problemsp. 262
Historical Note: Fourierp. 267
Historical Note: Dirichletp. 268
Anatomy of an Application: Some Ideas from Quantum Mechanicsp. 270
Problems for Review and Discoveryp. 273
Laplace Transformsp. 277
Introductionp. 278
Applications to Differential Equationsp. 280
Derivatives and Integrals of Laplace Transformsp. 285
Convolutionsp. 291
Abel's Mechanical Problemp. 293
The Unit Step and Impulse Functionsp. 298
Historical Note: Laplacep. 305
Anatomy of an Application: Flow Initiated by an Impulsively Started Flat Platep. 306
Problems for Review and Discoveryp. 309
The Calculus of Variationsp. 315
Introductory Remarksp. 316
Euler's Equationp. 319
Isoperimetric Problems and the Likep. 327
Lagrange Multipliersp. 328
Integral Side Conditionsp. 329
Finite Side Conditionsp. 333
Historical Note: Newtonp. 338
Anatomy of an Application: Hamilton's Principle and its Implicationsp. 340
Problems for Review and Discoveryp. 344
Numerical Methodsp. 347
Introductory Remarksp. 348
The Method of Eulerp. 349
The Error Termp. 353
An Improved Euler Methodp. 357
The Runge-Kutta Methodp. 360
Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equationsp. 365
Problems for Review and Discoveryp. 368
Systems of First-Order Equationsp. 371
Introductory Remarksp. 372
Linear Systemsp. 374
Homogeneous Linear Systems with Constant Coefficientsp. 382
Nonlinear Systems: Volterra's Predator-Prey Equationsp. 389
Anatomy of an Application: Solution of Systems with Matrices and Exponentialsp. 395
Problems for Review and Discoveryp. 400
The Nonlinear Theoryp. 403
Some Motivating Examplesp. 404
Specializing Downp. 404
Types of Critical Points: Stabilityp. 409
Critical Points and Stability for Linear Systemsp. 417
Stability by Liapunov's Direct Methodp. 427
Simple Critical Points of Nonlinear Systemsp. 432
Nonlinear Mechanics: Conservative Systemsp. 439
Periodic Solutions: The Poincare-Bendixson Theoremp. 444
Historical Note: Poincarep. 452
Anatomy of an Application: Mechanical Analysis of a Block on a Springp. 454
Problems for Review and Discoveryp. 457
Dynamical Systemsp. 461
Flowsp. 462
Dynamical Systemsp. 464
Stable and Unstable Fixed Pointsp. 466
Linear Dynamics in the Planep. 468
Some Ideas from Topologyp. 475
Open and Closed Setsp. 475
The Idea of Connectednessp. 476
Closed Curves in the Planep. 478
Planar Autonomous Systemsp. 480
Ingredients of the Proof of Poincare-Bendixsonp. 480
Anatomy of an Application: Lagrange's Equationsp. 489
Problems for Review and Discoveryp. 493
Bibliographyp. 495
Answers to Odd-Numbered Exercisesp. 497
Indexp. 525
Table of Contents provided by Ingram. All Rights Reserved.

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