Differential Equations, 3rd Edition

Table of contents

1. Cover Page
2. Half-Title Page
3. Series Page
4. Title Page
5. Copyright Page
6. Dedication Page
7. Contents
8. Preface
9. Author
10. 1 What is a Differential Equation?
    1. 1.1 Introductory Remarks
    2. 1.2 A Taste of Ordinary Differential Equations
    3. 1.3 The Nature of Solutions
11. 2 Solving First-Order Equations
    1. 2.1 Separable Equations
    2. 2.2 First-Order Linear Equations
    3. 2.3 Exact Equations
    4. 2.4 Orthogonal Trajectories and Curves
    5. 2.5 Homogeneous Equations
    6. 2.6 Integrating Factors
    7. 2.7 Reduction of Order
       1. 2.7.1 Dependent Variable Missing
       2. 2.7.2 Independent Variable Missing
12. 3 Some Applications of the First-Order Theory
    1. 3.1 The Hanging Chain and Pursuit Curves
       1. 3.1.1 The Hanging Chain
       2. 3.1.2 Pursuit Curves
    2. 3.2 Electrical Circuits
13. 4 Second-Order Linear Equations
    1. 4.1 Second-Order Linear Equations with Constant Coefficients
    2. 4.2 The Method of Undetermined Coefficients
    3. 4.3 The Method of Variation of Parameters
    4. 4.4 The Use of a Known Solution to Find Another
    5. 4.5 Higher-Order Equations
14. 5 Applications of the Second-Order Theory
    1. 5.1 Vibrations and Oscillations
       1. 5.1.1 Undamped Simple Harmonic Motion
       2. 5.1.2 Damped Vibrations
       3. 5.1.3 Forced Vibrations
       4. 5.1.4 A Few Remarks about Electricity
    2. 5.2 Newton's Law of Gravitation and Kepler's Laws
       1. 5.2.1 Kepler's Second Law
       2. 5.2.2 Kepler's First Law
       3. 5.2.3 Kepler's Third Law
15. 6 Power Series Solutions and Special Functions
    1. 6.1 Introduction and Review of Power Series
       1. 6.1.1 Review of Power Series
    2. 6.2 Series Solutions of First-Order Equations
    3. 6.3 Ordinary Points
    4. 6.4 Regular Singular Points
    5. 6.5 More on Regular Singular Points
16. 7 Fourier Series: Basic Concepts
    1. 7.1 Fourier Coefficients
    2. 7.2 Some Remarks about Convergence
    3. 7.3 Even and Odd Functions: Cosine and Sine Series
    4. 7.4 Fourier Series on Arbitrary Intervals
    5. 7.5 Orthogonal Functions
17. 8 Laplace Transforms
    1. 8.0 Introduction
    2. 8.1 Applications to Differential Equations
    3. 8.2 Derivatives and Integrals
    4. 8.3 Convolutions
       1. 8.3.1 Abel's Mechanics Problem
    5. 8.4 The Unit Step and Impulse Functions
18. 9 The Calculus of Variations
    1. 9.1 Introductory Remarks
    2. 9.2 Euler's Equation
    3. 9.3 Isoperimetric Problems and the Like
       1. 9.3.1 Lagrange Multipliers
       2. 9.3.2 Integral Side Conditions
       3. 9.3.3 Finite Side Conditions
19. 10 Systems of First-Order Equations
    1. 10.1 Introductory Remarks
    2. 10.2 Linear Systems
    3. 10.3 Systems with Constant Coefficients
    4. 10.4 Nonlinear Systems
20. 11 Partial Differential Equations and Boundary Value Problems
    1. 11.1 Introduction and Historical Remarks
    2. 11.2 Eigenvalues and the Vibrating String
       1. 11.2.1 Boundary Value Problems
       2. 11.2.2 Derivation of the Wave Equation
       3. 11.2.3 Solution of the Wave Equation
    3. 11.3 The Heat Equation
    4. 11.4 The Dirichlet Problem for a Disc
       1. 11.4.1 The Poisson Integral
    5. 11.5 Sturm—Liouville Problems
21. 12 The Nonlinear Theory
    1. 12.1 Some Motivating Examples
    2. 12.2 Specializing Down
    3. 12.3 Types of Critical Points: Stability
    4. 12.4 Critical Points and Stability
    5. 12.5 Stability by Lyapunov's Direct Method
    6. 12.6 Simple Critical Points of Nonlinear Systems
    7. 12.7 Nonlinear Mechanics: Conservative Systems
    8. 12.8 Periodic Solutions
22. 13 Qualitative Properties and Theoretical Aspects
    1. 13.1 A Bit of Theory
    2. 13.2 Picard's Existence and Uniqueness Theorem
       1. 13.2.1 The Form of a Differential Equation
       2. 13.2.2 Picard's Iteration Technique
       3. 13.2.3 Some Illustrative Examples
       4. 13.2.4 Estimation of the Picard Iterates
    3. 13.3 Oscillations and the Sturm Separation Theorem
    4. 13.4 The Sturm Comparison Theorem
23. Appendix: Review of Linear Algebra
24. Bibliography
25. Index