Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations.
This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutions, and that should be of interest to the mathematics major, such as the theorems about the existence and uniqueness of solutions. The final chapters discuss the stability of critical points of plane autonomous systems and the results about the existence of periodic solutions of nonlinear equations.
This book is great use to mathematicians, physicists, and undergraduate students of engineering and the science who are interested in applications of differential equation.
Inhalt
Preface
1. Linear Differential Equations
1.1 Introduction
1.2 The Fundamental Theorem
1.3 First-Order Linear Equations
1.4 Linear Dependence
1.5 The Wronskian
1.6 Abel's Formula
1.7 Fundamental Sets of Solutions
1.8 Polynomial Operators
1.9 Equations with Constant Coefficients
1.10 Equations of Cauchy Type
1.11 The Nonhomogeneous Equation
1.12 Variation of Parameters
1.13 The Method of Undetermined Coefficients
1.14 Applications
References
2. Further Properties of Linear Differential Equations
2.1 Reduction of Order
2.2 Factorization of Operators
2.3 Some Variable Changes
2.4 Zeros of Solutions
References
3. Complex Variables
3.1 Introduction
3.2 Functions of a Complex Variable
3.3 Complex Series
3.4 Power Series
3.5 Taylor Series
References
4. Series Solutions
4.1 Introduction
4.2 Solutions at an Ordinary Point
4.3 Analyticity of Solutions at an Ordinary Point
4.4 Regular Singular Points
4.5 Solutions at a Regular Singular Point
4.6 The Method of Frobenius
4.7 The Case of Equal Exponents
4.8 When the Exponents Differ by a Positive Integer
4.9 Complex Exponents
4.10 The Point at Infinity
References
5. Bessel Functions
5.1 The Gamma Function
5.2 Bessel's Equation
5.3 Bessel Functions of the Second and Third Kinds
5.4 Properties of Bessel Functions
5.5 Modified Bessel Functions
5.6 Other Forms for Bessel's Functions
References
6. Orthogonal Polynomials
6.1 Orthogonal Functions
6.2 An Existence Theorem for Orthogonal Polynomials
6.3 Some Properties of Orthogonal Polynomials
6.4 Generating Functions
6.5 Legendre Polynomials
6.6 Properties of Legendre Polynomials
6.7 Orthogonality
6.8 Legendre's Differential Equation
6.9 Tchebycheff Polynomials
6.10 Other Sets of Orthogonal Polynomials
References
7. Eigenvalue Problems
7.1 Introduction
7.2 The Adjoint Equation
7.3 Boundary Operators
7.4 Self-Adjoint Eigenvalue Problems
7.5 Properties of Self-Adjoint Problems
7.6 Some Special Types of Self-Adjoint Problems
7.7 Singular Problems
7.8 Some Important Singular Problems
References
8. Fourier Series
8.1 Orthogonal Sets of Functions
8.2 Fourier Series
8.3 Examples of Fourier Series
8.4 Types of Convergence
8.5 Convergence in the Mean
8.6 Closed Orthogonal Sets
8.7 Pointwise Convergence of the Trigonometric Series
8.8 The Sine and Cosine Series
8.9 Other Fourier Series
References
9. Systems of Differential Equations
9.1 First-Order Systems
9.2 Systems with Constant Coefficients
9.3 Applications
References
10. Laplace Transforms
10.1 The Laplace Transform
10.2 Conditions for the Existence of the Laplace Transform
10.3 Properties of Laplace Transforms
10.4 Inverse Transforms
10.5 Application to Differential Equations
References
11. Partial Differential Equations and Boundary-Value Problems
11.1 Introduction
11.2 The Heat Equation
11.3 The Method of Separation of Variables
11.4 Steady State Heat Flow
11.5 The Vibrating String
11.6 The Solution of the Problem of the Vibrating String
11.7 The Laplacian in Other Coordinate Systems
11.8 A Problem in Cylindrical Coordinates
11.9 A Problem in Spherical Coordinates
11.10 Double Fourier Series
References
12. Nonlinear Differential Equations
12.1 First-Order Equations
12.2 Exact Equations
12.3 Some Special Types of Second-Order Equations
12.4 Existence and Uniqueness of Solutions
12.5 Existence and Uniqueness of Solutions for Systems
12.6 The Phase Plane
12.7 Critical Points
12.8 Stability for Nonlinear Systems
12.9 Perturbed Linear Systems
12.10 Periodic Solutions
References
Appendix
Answers to Miscellaneous Exercises
Subject Index