UID:
almahu_9949232518802882
Format:
1 online resource (882 pages) :
,
illustrations
Edition:
Fourth edition.
ISBN:
0-12-804777-1
Note:
Front Cover -- Differential Equations with Mathematica -- Copyright -- Contents -- Preface -- Chapter 1: Introduction to Differential Equations -- 1.1 Definitions and Concepts -- 1.2 Solutions of Differential Equations -- 1.3 Initial and Boundary-Value Problems -- 1.4 Direction Fields -- 1.4.1 Creating Interactive Applications -- Chapter 2: First-Order Ordinary Differential Equations -- 2.1 Theory of First-Order Equations: A Brief Discussion -- 2.2 Separation of Variables -- Application: Kidney Dialysis -- 2.3 Homogeneous Equations -- Application: Models of Pursuit -- 2.4 Exact Equations -- 2.5 Linear Equations -- 2.5.1 Integrating Factor Approach -- 2.5.2 Variation of Parameters and the Method of Undetermined Coefficients -- Application: Antibiotic Production -- 2.6 Numerical Approximations of Solutions to First-Order Equations -- 2.6.1 Built-In Methods -- Application: Modeling the Spread of a Disease -- 2.6.2 Other Numerical Methods -- Euler's Method -- Improved Euler's Method -- The Runge-Kutta Method -- Chapter 3: Applications of First-Order Equations -- 3.1 Orthogonal Trajectories -- Application: Oblique Trajectories -- 3.2 Population Growth and Decay -- 3.2.1 The Malthus Model -- 3.2.2 The Logistic Equation -- Application: Harvesting -- Application: The Logistic Difference Equation -- 3.3 Newton's Law of Cooling -- 3.4 Free-Falling Bodies -- Chapter 4: Higher-Order Differential Equations -- 4.1 Preliminary Definitions and Notation -- 4.1.1 Introduction -- 4.1.2 The nth-Order Ordinary Linear Differential Equation -- 4.1.3 Fundamental Set of Solutions -- 4.1.4 Existence of a Fundamental Set of Solutions -- 4.1.5 Reduction of Order -- 4.2 Solving Homogeneous Equations With Constant Coefficients -- 4.2.1 Second-Order Equations -- 4.2.2 Higher-Order Equations.
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4.3 Introduction to Solving Nonhomogeneous Equations -- 4.4 Nonhomogeneous Equations With Constant Coefficients: The Method of Undetermined Coefficients -- Outline of the Method of Undetermined Coefficients -- Determining the Form of yp(x) (Step 2): -- 4.4.1 Second-Order Equations -- 4.4.2 Higher-Order Equations -- 4.5 Nonhomogeneous Equations With Constant Coefficients: Variation of Parameters -- 4.5.1 Second-Order Equations -- Summary of Variation of Parameters for Second-Order Equations -- 4.5.2 Higher-Order Nonhomogeneous Equations -- 4.6 Cauchy-Euler Equations -- 4.6.1 Second-Order Cauchy-Euler Equations -- 4.6.2 Higher-Order Cauchy-Euler Equations -- 4.6.3 Variation of Parameters -- 4.7 Series Solutions -- 4.7.1 Power Series Solutions About Ordinary Points -- Power Series Solution Method About an Ordinary Point -- 4.7.2 Series Solutions About Regular Singular Points -- 4.7.3 Method of Frobenius -- Application: Zeros of the Bessel Functions of the First Kind -- Application: The Wave Equation on a Circular Plate -- 4.8 Nonlinear Equations -- Chapter 5: Applications of Higher-Order Differential Equations -- 5.1 Harmonic Motion -- 5.1.1 Simple Harmonic Motion -- 5.1.2 Damped Motion -- 5.1.3 Forced Motion -- 5.1.4 Soft Springs -- 5.1.5 Hard Springs -- 5.1.6 Aging Springs -- Application: Hearing Beats and Resonance -- 5.2 The Pendulum Problem -- 5.3 Other Applications -- 5.3.1 L-R-C Circuits -- 5.3.2 Deflection of a Beam -- 5.3.3 Bodé Plots -- 5.3.4 The Catenary -- Chapter 6: Systems of Ordinary Differential Equations -- 6.1 Review of Matrix Algebra and Calculus -- 6.1.1 Defining Nested Lists, Matrices, and Vectors -- 6.1.2 Extracting Elements of Matrices -- 6.1.3 Basic Computations With Matrices -- 6.1.4 Systems of Linear Equations -- 6.1.5 Eigenvalues and Eigenvectors -- 6.1.6 Matrix Calculus.
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6.2 Systems of Equations: Preliminary Definitions and Theory -- 6.2.1 Preliminary Theory -- 6.2.2 Linear Systems -- 6.3 Homogeneous Linear Systems With Constant Coefficients -- 6.3.1 Distinct Real Eigenvalues -- 6.3.2 Complex Conjugate Eigenvalues -- 6.3.3 Alternate Method for Solving Initial-Value Problems -- 6.3.4 Repeated Eigenvalues -- 6.4 Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential -- 6.4.1 Undetermined Coefficients -- 6.4.2 Variation of Parameters -- 6.4.3 The Matrix Exponential -- 6.5 Numerical Methods -- 6.5.1 Built-In Methods -- Application: Controlling the Spread of a Disease -- 6.5.2 Euler's Method -- 6.5.3 Runge-Kutta Method -- 6.6 Nonlinear Systems, Linearization, and Classification of Equilibrium Points -- 6.6.1 Real Distinct Eigenvalues -- 6.6.2 Repeated Eigenvalues -- 6.6.3 Complex Conjugate Eigenvalues -- 6.6.4 Nonlinear Systems -- Classification of Equilibrium Points -- Chapter 7: Applications of Systems of Ordinary Differential Equations -- 7.1 Mechanical and Electrical Problems With First-Order Linear Systems -- 7.1.1 L-R-C Circuits With Loops -- 7.1.2 L-R-C Circuit With One Loop -- 7.1.3 L-R-C Circuit With Two Loops -- 7.1.4 Spring-Mass Systems -- 7.2 Diffusion and Population Problems With First-Order Linear Systems -- 7.2.1 Diffusion Through a Membrane -- 7.2.2 Diffusion Through a Double-Walled Membrane -- 7.2.3 Population Problems -- 7.3 Applications That Lead to Nonlinear Systems -- 7.3.1 Biological Systems: Predator-Prey Interactions, The Lotka-Volterra System, and Food Chains in the Chemostat -- The Lotka-Volterra System -- Simple Food Chain in a Chemostat -- Long Food Chain in a Chemostat -- 7.3.2 Physical Systems: Variable Damping -- 7.3.3 Differential Geometry: Curvature -- Chapter 8: Laplace Transform Methods.
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8.1 The Laplace Transform -- 8.1.1 Definition of the Laplace Transform -- 8.1.2 Exponential Order, Jump Discontinuities and Piecewise-Continuous Functions -- 8.1.3 Properties of the Laplace Transform -- 8.2 The Inverse Laplace Transform -- 8.2.1 Definition of the Inverse Laplace Transform -- Linear Factors (Nonrepeated) -- Repeated Linear Factors -- Irreducible Quadratic Factors -- 8.2.2 Laplace Transform of an Integral -- 8.3 Solving Initial-Value Problems With the Laplace Transform -- 8.4 Laplace Transforms of Step and Periodic Functions -- 8.4.1 Piecewise-Defined Functions: The Unit Step Function -- 8.4.2 Solving Initial-Value Problems With Piecewise-Continuous Forcing Functions -- 8.4.3 Periodic Functions -- 8.4.4 Impulse Functions: The Delta Function -- 8.5 The Convolution Theorem -- 8.5.1 The Convolution Theorem -- 8.5.2 Integral and Integrodifferential Equations -- 8.6 Applications of Laplace Transforms, Part I -- 8.6.1 Spring-Mass Systems Revisited -- 8.6.2 L-R-C Circuits Revisited -- 8.6.3 Population Problems Revisited -- Application: The Tautochrone -- 8.7 Laplace Transform Methods for Systems -- 8.8 Applications of Laplace Transforms, Part II -- 8.8.1 Coupled Spring-Mass Systems -- 8.8.2 The Double Pendulum -- Application: Free Vibration of a Three-Story Building -- Chapter 9: Eigenvalue Problems and Fourier Series -- 9.1 Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems -- 9.1.1 Boundary-Value Problems -- 9.1.2 Eigenvalue Problems -- 9.1.3 Sturm-Liouville Problems -- 9.2 Fourier Sine Series and Cosine Series -- 9.2.1 Fourier Sine Series -- 9.2.2 Fourier Cosine Series -- 9.3 Fourier Series -- 9.3.1 Fourier Series -- 9.3.2 Even, Odd, and Periodic Extensions -- 9.3.3 Differentiation and Integration of Fourier Series -- 9.3.4 Parseval's Equality -- 9.4 Generalized Fourier Series.
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Chapter 10: Partial Differential Equations -- 10.1 Introduction to Partial Differential Equations and Separation of Variables -- 10.1.1 Introduction -- 10.1.2 Separation of Variables -- 10.2 The One-Dimensional Heat Equation -- 10.2.1 The Heat Equation With Homogeneous Boundary Conditions -- 10.2.2 Nonhomogeneous Boundary Conditions -- 10.2.3 Insulated Boundary -- 10.3 The One-Dimensional Wave Equation -- 10.3.1 The Wave Equation -- 10.3.2 D'Alembert's Solution -- 10.4 Problems in Two Dimensions: Laplace's Equation -- 10.4.1 Laplace's Equation -- 10.5 Two-Dimensional Problems in a Circular Region -- 10.5.1 Laplace's Equation in a Circular Region -- 10.5.2 The Wave Equation in a Circular Region -- Appendix A: Appendix Getting Started -- Introduction to Mathematica -- A Note Regarding Different Versions of Mathematica -- Getting Started With Mathematica -- Five Basic Rules of Mathematica Syntax -- Getting Help From Mathematica -- Mathematica Help -- The Mathematica Menu -- Bibliography -- Index -- Back Cover.
Additional Edition:
ISBN 0-12-804776-3
Language:
English
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