Differential Equations with Boundary Value Problems

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  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2005-07-28
  • Publisher: Pearson
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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

Prefacep. ix
Introduction to Differential Equationsp. 1
Differential Equation Modelsp. 2
The Derivativep. 7
Integrationp. 10
First-Order Equationsp. 18
Differential Equations and Solutionsp. 18
Solutions to Separable Equationsp. 31
Models of Motionp. 44
Linear Equationsp. 54
Mixing Problemsp. 64
Exact Differential Equationsp. 73
Existence and Uniqueness of Solutionsp. 90
Dependence of Solutions on Initial Conditionsp. 102
Autonomous Equations and Stabilityp. 107
The Daredevil Skydiverp. 120
Modeling and Applicationsp. 123
Modeling Population Growthp. 124
Models and the Real Worldp. 138
Personal Financep. 143
Electrical Circuitsp. 152
The Spruce Budwormp. 158
Social Security, Now or Laterp. 161
Second-Order Equationsp. 163
Definitions and Examplesp. 163
Second-Order Equations and Systemsp. 174
Linear, Homogeneous Equations with Constant Coefficientsp. 179
Harmonic Motionp. 190
Inhomogeneous Equations; the Method of Undetermined Coefficientsp. 199
Variation of Parametersp. 209
Forced Harmonic Motionp. 215
Nonlinear Oscillatorsp. 228
The Laplace Transformp. 231
The Definition of the Laplace Transformp. 232
Basic Properties of the Laplace Transformp. 241
The Inverse Laplace Transformp. 248
Using the Laplace Transform to Solve Differential Equationsp. 256
Discontinuous Forcing Termsp. 266
The Delta Functionp. 280
Convolutionsp. 287
Summaryp. 298
Forced Harmonic Oscillatorsp. 299
Numerical Methodsp. 301
Euler's Methodp. 302
Runge-Kutta Methodsp. 314
Numerical Error Comparisonsp. 322
Practical Use of Solversp. 327
A Cautionary Talep. 332
Numerical Error Comparisonp. 334
Matrix Algebrap. 335
Vectors and Matricesp. 335
The Geometry of Systems of Linear Equationsp. 348
Solving Systems of Equationsp. 353
Properties of Solution Setsp. 362
Subspacesp. 371
Determinantsp. 384
An Introduction to Systemsp. 396
Definitions and Examplesp. 396
Geometric Interpretation of Solutionsp. 405
Qualitative Analysisp. 417
Linear Systemsp. 425
Long-Term Behavior of Solutionsp. 441
Linear Systems with Constant Coefficientsp. 444
Overview of the Techniquep. 444
Planar Systemsp. 452
Phase Plane Portraitsp. 466
Higher Dimensional Systemsp. 484
The Exponential of a Matrixp. 492
Qualitative Analysis of Linear Systemsp. 510
Higher-Order Linear Equationsp. 515
Inhomogeneous Linear Systemsp. 528
Phase Plane Portraitsp. 538
Oscillations of Linear Moleculesp. 539
Nonlinear Systemsp. 545
The Linearization of a Nonlinear Systemp. 545
Long-Term Behavior of Solutionsp. 559
Invariant Sets and the Use of Nullclinesp. 566
Long-Term Behavior of Solutions to Planar Systemsp. 574
Conserved Quantitiesp. 586
Nonlinear Mechanicsp. 592
The Method of Lyapunovp. 609
Predator-Prey Systemsp. 619
Human Immune Response to Infectious Diseasep. 631
Analysis of Competing Speciesp. 634
Series Solutions to Differential Equationsp. 637
Review of Power Seriesp. 638
Series Solutions Near Ordinary Pointsp. 650
Legendre's Equationp. 662
Types of Singular Points--Euler's Equationp. 668
Series Solutions Near Regular Singular Pointsp. 677
Solutions in the Exceptional Casesp. 690
Bessel's Equation and Bessel Functionsp. 701
Fourier Seriesp. 712
Computation of Fourier Seriesp. 713
Convergence of Fourier Seriesp. 724
Fourier Cosine and Sine Seriesp. 733
The Complex Form of a Fourier Seriesp. 738
The Discrete Fourier Transform and the FFTp. 741
Partial Differential Equationsp. 750
Derivation of the Heat Equationp. 750
Separation of Variables for the Heat Equationp. 756
The Wave Equationp. 768
Laplace's Equationp. 778
Laplace's Equation on a Diskp. 785
Sturm Liouville Problemsp. 791
Orthogonality and Generalized Fourier Seriesp. 801
Temperature in a Ball--Legendre Polynomialsp. 810
Time Dependent PDEs in Higher Dimensionp. 814
Domains with Circular Symmetry--Bessel Functionsp. 821
Answers to Odd-Numbered Problemsp. 1
Indexp. 1
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