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# Differential Equations with Applications and Historical Notes

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## Description

The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations. Specially designed for just such a course, Differential Equations with Applications and Historical Notes takes great pleasure in the journey into the world of differential equations and their wide range of applications. The author-a highly respected educator-advocates a careful approach, using explicit explanation to ensure students fully comprehend the subject matter.

With an emphasis on modeling and applications, the long-awaited Third Edition of this classic textbook presents a substantial new section on Gauss's bell curve and improves coverage of Fourier analysis, numerical methods, and linear algebra. Relating the development of mathematics to human activity-i.e., identifying why and how mathematics is used-the text includes a wealth of unique examples and exercises, as well as the author's distinctive historical notes, throughout.

Provides an ideal text for a one- or two-semester introductory course on differential equations

Emphasizes modeling and applications

Presents a substantial new section on Gauss's bell curve

Improves coverage of Fourier analysis, numerical methods, and linear algebra

Relates the development of mathematics to human activity-i.e., identifying why and how mathematics is used

Includes a wealth of unique examples and exercises, as well as the author's distinctive historical notes, throughout

Uses explicit explanation to ensure students fully comprehend the subject matter

Outstanding Academic Title of the Year, Choice magazine, American Library Association.

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## Product details

- Hardback | 764 pages
- 156 x 235 x 43.18mm | 1,225g
- 25 Oct 2016
- Taylor & Francis Inc
- Productivity Press
- Portland, United States
- English
- New edition
- 3rd New edition
- 17 Tables, black and white; 102 Illustrations, black and white
- 1498702597
- 9781498702591
- 1,033,767

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## Table of contents

Introduction

General Remarks on Solutions

Families of Curves: Orthogonal Trajectories

Growth, Decay, Chemical Reactions, and Mixing

Falling Bodies and Other Motion Problems

Brachistochrone: Fermat and the Bernoullis

Miscellaneous Problems for Chapter 1

Appendix: Some Ideas from the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation

First-Order Equations

Homogeneous Equations

Exact Equations

Integrating Factors

Linear Equations

Reduction of Order

Hanging Chain: Pursuit Curves

Simple Electric Circuits

Miscellaneous Problems for Chapter 2

Second-Order Linear Equations

Introduction

General Solution of the Homogeneous Equation

Use of a Known Solution to Find Another

Homogeneous Equation with Constant Coefficients

Method of Undetermined Coefficients

Method of Variation of Parameters

Vibrations in Mechanical and Electrical Systems

Newton's Law of Gravitation and the Motion of the Planets

Higher-Order Linear Equations: Coupled Harmonic Oscillators

Operator Methods for Finding Particular Solutions

Appendix: Euler

Appendix: Newton

Qualitative Properties of Solutions

Oscillations and the Sturm Separation Theorem

Sturm Comparison Theorem

Power Series Solutions and Special Functions

Introduction: A Review of Power Series

Series Solutions of First-Order Equations

Second-Order Linear Equations: Ordinary Points

Regular Singular Points

Regular Singular Points (Continued)

Gauss's Hypergeometric Equation

Point at Infinity

Appendix: Two Convergence Proofs

Appendix: Hermite Polynomials and Quantum Mechanics

Appendix: Gauss

Appendix: Chebyshev Polynomials and the Minimax Property

Appendix: Riemann's Equation

Fourier Series and Orthogonal Functions

Fourier Coefficients

Problem of Convergence

Even and Odd Functions: Cosine and Sine Series

Extension to Arbitrary Intervals

Orthogonal Functions

Mean Convergence of Fourier Series

Appendix: A Pointwise Convergence Theorem

Partial Differential Equations and Boundary Value Problems

Introduction: Historical Remarks

Eigenvalues, Eigenfunctions, and the Vibrating String

Heat Equation

Dirichlet Problem for a Circle: Poisson's Integral

Sturm-Liouville Problems

Appendix: Existence of Eigenvalues and Eigenfunctions

Some Special Functions of Mathematical Physics

Legendre Polynomials

Properties of Legendre Polynomials

Bessel Functions: The Gamma Function

Properties of Bessel Functions

Appendix: Legendre Polynomials and Potential Theory

Appendix: Bessel Functions and the Vibrating Membrane

Appendix: Additional Properties of Bessel Functions

Laplace Transforms

Introduction

Few Remarks on the Theory

Applications to Differential Equations

Derivatives and Integrals of Laplace Transforms

Convolutions and Abel's Mechanical Problem

More about Convolutions: The Unit Step and Impulse

Functions

Appendix: Laplace

Appendix: Abel

Systems of First-Order Equations

General Remarks on Systems

Linear Systems

Homogeneous Linear Systems with Constant Coefficients

Nonlinear Systems: Volterra's Prey-Predator Equations

Nonlinear Equations

Autonomous Systems: The Phase Plane and Its Phenomena

Types of Critical Points: Stability

Critical Points and Stability for Linear Systems

Stability by Liapunov's Direct Method

Simple Critical Points of Nonlinear Systems

Nonlinear Mechanics: Conservative Systems

Periodic Solutions: The Poincare-Bendixson Theorem

More about the Van Der Pol Equation

Appendix: Poincare

Appendix: Proof of Lienard's Theorem

Calculus of Variations

Introduction: Some Typical Problems of the Subject

Euler's Differential Equation for an Extremal

Isoperimetric Problems

Appendix: Lagrange

Appendix: Hamilton's Principle and Its Implications

The Existence and Uniqueness of Solutions

Method of Successive Approximations

Picard's Theorem

Systems: Second-Order Linear Equation

Numerical Methods

(by John S. Robertson)

Introduction

Method of Euler

Errors

An Improvement to Euler

Higher-Order Methods

Systems

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## Review quote

--M. Bona, University of Florida

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## About George F. Simmons

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