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Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one's own time. An unfortunate effect of the predominance of fads is that if a student doesn't learn about such worthwhile topics as the wave equation, Gauss's hypergeometric function, the gamma function, and the basic problems of the calculus of variations-among others-as an undergraduate, then he/she is unlikely to do so later.

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.

Table of Contents

The Nature of Differential Equations: Separable Equations
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 Poincaré-Bendixson Theorem
More about the Van Der Pol Equation
Appendix: Poincaré
Appendix: Proof of Liénard'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