Differential Equations

Preface

1. What Is a Differential Equation?
1.1 Introductory Remarks
1.2 A Taste of Ordinary Differential Equations
1.3 The Nature of Solutions

2. Solving First-Order Equations
2.1 Separable Equations
2.2 First-Order Linear Equations
2.3 Exact Equations
2.4 Orthogonal Trajectories and Curves
2.5 Homogeneous Equations
2.6 Integrating Factors
2.7 Reduction of Order
2.7.1 Dependent Variable Missing
2.7.2 Independent Variable Missing

3. Some Applications of the First-Order Theory
3.1 The Hanging Chain and Pursuit Curves
3.1.1 The Hanging Chain
3.1.2 Pursuit Curves
3.2 Electrical Circuits
Anatomy of an Application
Problems for Review and Discovery

4. Second-Order Linear Equations
4.1 Second-Order Linear Equations with Constant Coefficients
4.2 The Method of Undetermined Coefficients
4.3 The Method of Variation of Parameters
4.4 The Use of a Known Solution to Find Another
4.5 Higher-Order Equations

5. Applications of the Second-Order Theory
5.1 Vibrations and Oscillations
5.1.1 Undamped Simple Harmonic Motion
5.1.2 Damped Vibrations
5.1.3 Forced Vibrations
5.1.4 A Few Remarks About Electricity
5.2 Newton's Law of Gravitation and Kepler's Laws
5.2.1 Kepler's Second Law
5.2.2 Kepler's First Law
5.2.3 Kepler's Third Law
Historical Note
Anatomy of an Application
Problems for Review and Discovery

6. Power Series Solutions and Special Functions
6.1 Introduction and Review of Power Series
6.1.1 Review of Power Series
6.2 Series Solutions of First-Order Equations
6.3 Ordinary Points
6.4 Regular Singular Points
6.5 More on Regular Singular Points
Historical Note
Historical Note
Anatomy of an Application
Problems for Review and Discovery

7. Fourier Series: Basic Concepts
7.1 Fourier Coefficients
7.2 Some Remarks about Convergence
7.3 Even and Odd Functions: Cosine and Sine Series
7.4 Fourier Series on Arbitrary Intervals
7.5 Orthogonal Functions
Historical Note
Anatomy of an Application
Problems for Review and Discovery

8. Laplace Transforms
8.0 Introduction
8.1 Applications to Differential Equations
8.2 Derivatives and Integrals
8.3 Convolutions
8.3.1 Abel's Mechanics Problem
8.4 The Unit Step and Impulse Functions
Historical Note
Anatomy of an Application
Problems for Review and Discovery

9. The Calculus of Variations
9.1 Introductory Remarks
9.2 Euler's Equation
9.3 Isoperimetric Problems and the Like
9.3.1 Lagrange Multipliers
9.3.2 Integral Side Conditions
9.3.3 Finite Side Conditions
Historical Note
Anatomy of an Application
Problems for Review and Discovery

10. Systems of First-Order Equations
10.1 Introductory Remarks
10.2 Linear Systems
10.3 Systems with Constant Coefficients
10.4 Nonlinear Systems
Anatomy of an Application
Problems for Review and Discovery

11. Partial Differential Equations and Boundary Value Problems
11.1 Introduction and Historical Remarks
11.2 Eigenvalues and the Vibrating String
11.2.1 Boundary Value Problems
11.2.2 Derivation of the Wave Equation
11.2.3 Solution of the Wave Equation
11.3 The Heat Equation
11.4 The Dirichlet Problem for a Disc
11.4.1 The Poisson Integral
11.5 Sturm-Liouville Problems
Historical Note
Historical Note
Anatomy of an Application
Problems for Review and Discovery

12. The Nonlinear Theory
12.1 Some Motivating Examples
12.2 Specializing Down
12.3 Types of Critical Points: Stability
12.4 Critical Points and Stability
12.5 Stability by Liapunov's Direct Method
12.6 Simple Critical Points of Nonlinear Systems
12.7 Nonlinear Mechanics: Conservative Systems
12.8 Periodic Solutions
Historical Note
Anatomy of an Application
Problems for Review and Discovery

13. Qualitative Properties and Theoretical Aspects
13.1 A Bit of Theory
13.2 Picard's Existence and Uniqueness Theorem
13.2.1 The Form of a Differential Equation
13.2.2 Picard's Iteration Technique
13.2.3 Some Illustrative Examples
13.2.4 Estimation of the Picard Iterates
13.3 Oscillations and the Sturm Separation Theorem
13.4 The Sturm Comparison Theorem
Anatomy of an Application
Problems for Review and Discovery

Appendix: Review of Linear Algebra
Bibliography
Index