Differential Equations with Boundary Value Problems
by Polking, John; Boggess, Al; Arnold, DavidEdition: 2nd
Format: Hardcover
Pub. Date: 2005-07-28
Publisher(s): Pearson
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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 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
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