Differential Equations with Applications and Historical Notes, Third Edition: (Textbooks in Mathematics)

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. A solutions manual is available upon qualifying course adoption. 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 Solutions manual available upon qualifying course adoption

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

About the Author :
George F. Simmons has academic degrees from the California Institute of Technology, Pasadena, California; the University of Chicago, Chicago, Illinois; and Yale University, New Haven, Connecticut. He taught at several colleges and universities before joining the faculty of Colorado College, Colorado Springs, Colorado, in 1962, where he is currently a professor of mathematics. In addition to Differential Equations with Applications and Historical Notes, Third Edition (CRC Press, 2016), Professor Simmons is the author of Introduction to Topology and Modern Analysis (McGraw-Hill, 1963), Precalculus Mathematics in a Nutshell (Janson Publications, 1981), and Calculus with Analytic Geometry (McGraw-Hill, 1985).