Introduction to Ordinary Differential Equations

Introduction to Ordinary Differential Equations

Second Enlarged Edition with Applications

2nd Edition - January 1, 1972

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  • Author: Albert L. Rabenstein
  • eBook ISBN: 9781483263854

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Introduction to Ordinary Differential Equations, Second Edition provides an introduction to differential equations. This book presents the application and includes problems in chemistry, biology, economics, mechanics, and electric circuits. Organized into 12 chapters, this edition begins with an overview of the methods for solving single differential equations. This text then describes the important basic properties of solutions of linear differential equations and explains higher-order linear equations. Other chapters consider the possibility of representing the solutions of certain linear differential equations in terms of power series. This book discusses as well the important properties of the gamma function and explains the stability of solutions and the existence of periodic solutions. The final chapter deals with the method for the construction of a solution of the integral equation and explains how to establish the existence of a solution of the initial value system. This book is a valuable resource for mathematicians, students, and research workers.

Table of Contents

  • Preface

    I Introduction to Differential Equations

    1.1 Introduction

    1.2 Separable Equations

    1.3 Exact Equations

    1.4 First-Order Linear Equations

    1.5 Orthogonal Trajectories

    1.6 Decay and Mixing Problems

    1.7 Population Growth

    1.8 An Economic Model

    1.9 Cooling: The Rate of a Chemical Reaction

    1.10 Two Special Types of Second-Order Equations

    1.11 Falling Bodies


    II Linear Differential Equations

    2.1 Introduction

    2.2 Linear Dependence

    2.3 Wronskians

    2.4 Polynomial Operators

    2.5 Complex Solutions

    2.6 Equations with Constant Coefficients

    2.7 Cauchy-Euler Equations

    2.8 Nonhomogeneous Equations

    2.9 The Method of Undetermined Coefficients

    2.10 Variation of Parameters

    2.11 Simple Harmonic Motion

    2.12 Electric Circuits

    2.13 Theory of Linear Equations


    III Series Solutions

    3.1 Power Series

    3.2 Taylor Series

    3.3 Ordinary Points

    3.4 Regular Singular Points

    3.5 The Case of Equal Exponents

    3.6 The Case When the Exponents Differ by an Integer

    3.7 The Point at Infinity

    3.8 Convergence of the Series


    IV Bessel Functions

    4.1 The Gamma Function

    4.2 Bessel's Equation

    4.3 Bessel Functions of the Second and Third Kinds

    4.4 Properties of Bessel Functions

    4.5 Modified Bessel Functions

    4.6 Other Forms for Bessel's Equation


    V Orthogonal Polynomials

    5.1 Orthogonal Functions

    5.2 An Existence Theorem for Orthogonal Polynomials

    5.3 Properties of Orthogonal Polynomials

    5.4 Generating Functions

    5.5 Legendre Polynomials

    5.6 Properties of Legendre Polynomials

    5.7 Orthogonality

    5.8 Legendre's Differential Equation

    5.9 Tchebycheff Polynomials

    5.10 Other Sets of Orthogonal Polynomials

    Table of Orthogonal Polynomials


    VI Eigenvalue Problems

    6.1 Introduction

    6.2 Self-Adjoint Problems

    6.3 Some Special Cases

    6.4 Singular Problems

    6.5 Some Important Singular Problems


    VII Fourier Series

    7.1 Introduction

    7.2 Examples of Fourier Series

    7.3 Types of Convergence

    7.4 Convergence in the Mean

    7.5 Complete Orthogonal Sets

    7.6 Pointwise Convergence of the Trigonometric Series

    7.7 The Sine and Cosine Series

    7.8 Other Fourier Series


    VIII Systems of Differential Equations

    8.1 Introduction

    8.2 First-Order Systems

    8.3 Linear Systems with Constant Coefficients

    8.4 Mechanical Systems

    8.5 Electric Circuits

    8.6 Some Problems from Biology


    IX Laplace Transforms

    9.1 The Laplace Transform

    9.2 Conditions for the Existence of the Laplace Transform

    9.3 Properties of Laplace Transforms

    9.4 Inverse Transforms

    9.5 Applications to Differential Equations


    X Partial Differential Equations and Boundary Value Problems

    10.1 Introduction

    10.2 The Heat Equation

    10.3 The Method of Separation of Variables

    10.4 Steady-State Heat Flow

    10.5 The Vibrating String

    10.6 The Solution of the Problem of the Vibrating String

    10.7 The Laplacian in Other Coordinate Systems

    10.8 A Problem in Cylindrical Coordinates

    10.9 A Problem in Spherical Coordinates

    10.10 Double Fourier Series


    XI The Phase Plane

    11.1 Introduction

    11.2 Stability

    11.3 The Method of Liapunov

    11.4 Perturbed Linear Systems

    11.5 Periodic Solutions


    XII Existence and Uniqueness of Solutions

    12.1 Preliminaries

    12.2 Successive Approximations

    12.3 Vector Functions

    12.4 First-Order Systems



    A1 Determinants

    A2 Properties of Determinants

    A3 Cofactors

    A4 Cramer's Rule

    Answers to Selected Exercises

    Subject Index

Product details

  • No. of pages: 536
  • Language: English
  • Copyright: © Academic Press 1972
  • Published: January 1, 1972
  • Imprint: Academic Press
  • eBook ISBN: 9781483263854

About the Author

Albert L. Rabenstein

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