Differential Equations: Theory, Technique, and Practice / Edition 1

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This traditional text is intended for mainstream one- or two-semester differential equations courses taken by undergraduates majoring in engineering, mathematics, and the sciences. Written by two of the world’s leading authorities on differential equations, Simmons/Krantz provides a cogent and accessible introduction to ordinary differential equations written in classical style. Its rich variety of modern applications in engineering, physics, and the applied sciences illuminate the concepts and techniques that students will use through practice to solve real-life problems in their careers.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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Table of Contents

Preface     VII
What is a Differential Equation?     1
Introductory Remarks     2
The Nature of Solutions     4
Separable Equations     10
First-Order Linear Equations     13
Exact Equations     17
Orthogonal Trajectories and Families of Curves     22
Homogeneous Equations     26
Integrating Factors     29
Reduction of Order     33
Dependent Variable Missing     33
Independent Variable Missing     35
The Hanging Chain and Pursuit Curves     38
The Hanging Chain     38
Pursuit Curves     42
Electrical Circuits     45
Anatomy of an Application: The Design of a Dialysis Machine     49
Problems for Review and Discovery     53
Second-Order Linear Equations     57
Second-Order Linear Equations with Constant Coefficients     58
The Method of Undetermined Coefficients     63
The Method of Variation of Parameters     67
The Use of a Known Solution to Find Another     71
Vibrations and Oscillations     75
Undamped Simple Harmonic Motion     75
Damped Vibrations     77
Forced Vibrations     80
A Few Remarks about Electricity     82
Newton's Law of Gravitation and Kepler's Laws     84
Kepler's Second Law     86
Kepler's First Law     87
Kepler's Third Law     89
Higher Order Linear Equations, Coupled Harmonic Oscillators     93
Historical Note: Euler     99
Anatomy of an Application: Bessel Functions and the Vibrating Membrane     101
Problems for Review and Discovery     105
Qualitative Properties and Theoretical Aspects     109
Review of Linear Algebra     110
Vector Spaces     110
The Concept of Linear Independence     111
Bases     113
Inner Product Spaces     114
Linear Transformations and Matrices     115
Eigenvalues and Eigenvectors     117
A Bit of Theory     119
Picard's Existence and Uniqueness Theorem     125
The Form of a Differential Equation     125
Picard's Iteration Technique     126
Some Illustrative Examples     127
Estimation of the Picard Iterates     129
Oscillations and the Sturm Separation Theorem     130
The Sturm Comparison Theorem      138
Anatomy of an Application: The Green's Function     142
Problems for Review and Discovery     146
Power Series Solutions and Special Functions     149
Introduction and Review of Power Series     150
Review of Power Series     150
Series Solutions of First-Order Differential Equations     159
Second-Order Linear Equations: Ordinary Points     164
Regular Singular Points     171
More on Regular Singular Points     177
Gauss's Hypergeometric Equation     184
Historical Note: Gauss     189
Historical Note: Abel     190
Anatomy of an Application: Steady-State Temperature in a Ball     192
Problems for Review and Discovery     194
Fourier Series: Basic Concepts     197
Fourier Coefficients     198
Some Remarks about Convergence     207
Even and Odd Functions: Cosine and Sine Series     211
Fourier Series on Arbitrary Intervals     218
Orthogonal Functions     221
Historical Note: Riemann     225
Anatomy of an Application: Introduction to the Fourier Transform     227
Problems for Review and Discovery     236
Partial Differential Equations and Boundary Value Problems     239
Introduction and Historical Remarks     240
Eigenvalues, Eigenfunctions, and the Vibrating String     243
Boundary Value Problems     243
Derivation of the Wave Equation     244
Solution of the Wave Equation     246
The Heat Equation     251
The Dirichlet Problem for a Disc     256
The Poisson Integral     259
Sturm-Liouville Problems     262
Historical Note: Fourier     267
Historical Note: Dirichlet     268
Anatomy of an Application: Some Ideas from Quantum Mechanics     270
Problems for Review and Discovery     273
Laplace Transforms     277
Introduction     278
Applications to Differential Equations     280
Derivatives and Integrals of Laplace Transforms     285
Convolutions     291
Abel's Mechanical Problem     293
The Unit Step and Impulse Functions     298
Historical Note: Laplace     305
Anatomy of an Application: Flow Initiated by an Impulsively Started Flat Plate     306
Problems for Review and Discovery     309
The Calculus of Variations     315
Introductory Remarks     316
Euler's Equation     319
Isoperimetric Problems and the Like     327
Lagrange Multipliers     328
Integral Side Conditions     329
Finite Side Conditions     333
Historical Note: Newton     338
Anatomy of an Application: Hamilton's Principle and its Implications     340
Problems for Review and Discovery     344
Numerical Methods     347
Introductory Remarks     348
The Method of Euler     349
The Error Term     353
An Improved Euler Method     357
The Runge-Kutta Method     360
Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations     365
Problems for Review and Discovery     368
Systems of First-Order Equations     371
Introductory Remarks     372
Linear Systems     374
Homogeneous Linear Systems with Constant Coefficients     382
Nonlinear Systems: Volterra's Predator-Prey Equations     389
Anatomy of an Application: Solution of Systems with Matrices and Exponentials     395
Problems for Review and Discovery     400
The Nonlinear Theory     403
Some Motivating Examples     404
Specializing Down     404
Types of Critical Points: Stability     409
Critical Points and Stability for Linear Systems     417
Stability by Liapunov's Direct Method     427
Simple Critical Points of Nonlinear Systems     432
Nonlinear Mechanics: Conservative Systems     439
Periodic Solutions: The Poincare-Bendixson Theorem     444
Historical Note: Poincare     452
Anatomy of an Application: Mechanical Analysis of a Block on a Spring     454
Problems for Review and Discovery     457
Dynamical Systems     461
Flows     462
Dynamical Systems     464
Stable and Unstable Fixed Points     466
Linear Dynamics in the Plane     468
Some Ideas from Topology     475
Open and Closed Sets     475
The Idea of Connectedness     476
Closed Curves in the Plane     478
Planar Autonomous Systems     480
Ingredients of the Proof of Poincare-Bendixson     480
Anatomy of an Application: Lagrange's Equations     489
Problems for Review and Discovery     493
Bibliography     495
Answers to Odd-Numbered Exercises     497
Index     525
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