Differential Equations with Applications and Historical Notes

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CRC Press, 2016 M11 17 - 764 páginas

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.

 

Páginas seleccionadas

Contenido

Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Suggestions for the Instructor
About the Author
i
The Nature of Differential Equations Separable Equations 1 Introduction
ii
General Remarks on Solutions
iii
Families of Curves Orthogonal Trajectories
ix
Orthogonal Functions
xi
The Mean Convergence of Fourier Series
xviii
Appendix A A Pointwise Convergence Theorem
xxiii
Partial Differential Equations and Boundary Value Problems 39 Introduction Historical Remarks
xxvii
Eigenvalues Eigenfunctions and the Vibrating String
i
The Heat Equation
viii
The Dirichlet Problem for a Circle Poissons Integral
xii
SturmLiouville Problems
xvi

Growth Decay Chemical Reactions and Mixing
vii
Falling Bodies and Other Motion Problems
xiv
The Brachistochrone Fermat and the Bernoullis
xix
Some Ideas From the Theory of Probability The Normal Distribution Curve or Bell Curve and Its Differential Equation
xxviii
First Order Equations 7 Homogeneous Equations
i
Exact Equations
iii
Integrating Factors
vi
Linear Equations
x
Reduction of Order
ix
The Hanging Chain Pursuit Curves
vii
Simple Electric Circuits
x
Second Order Linear Equations 14 Introduction
v
The General Solution of the Homogeneous Equation
ix
The Use of a Known Solution to find Another
i
The Homogeneous Equation with Constant Coefficients
iii
The Method of Undetermined Coefficients
vi
The Method of Variation of Parameters
x
Vibrations in Mechanical and Electrical Systems
vii
Newtons Law of Gravitation and The Motion of the Planets
iv
Higher Order Linear Equations Coupled Harmonic Oscillators
xcix
Operator Methods for Finding Particular Solutions
ciii
Appendix A Euler
cix
Appendix B Newton
cxv
Qualitative Properties of Solutions 24 Oscillations and the Sturm Separation Theorem
i
The Sturm Comparison Theorem
v
Power Series Solutions and Special Functions
viii
The Point at Infinity
iv
Appendix A Two Convergence Proofs
vi
Appendix B Hermite Polynomials and Quantum Mechanics
ix
Gauss
c
Chebyshev Polynomials and the Minimax Property
iv
Appendix E Riemanns Equation
viii
Fourier Series and Orthogonal Functions 33 The Fourier Coefficients
v
The Problem of Convergence
ix
Even and Odd Functions Cosine and Sine Series
ix
Extension to Arbitrary Intervals
viii
Appendix A The Existence of Eigenvalues and Eigenfunctions
xxii
Some Special Functions of Mathematical Physics 44 Legendre Polynomials
xxvi
Properties of Legendre Polynomials
2
Bessel Functions The Gamma Function
6
Properties of Bessel Functions
13
Appendix A Legendre Polynomials and Potential Theory
24
Appendix B Bessel Functions and the Vibrating Membrane
30
Additional Properties of Bessel Functions
33
Laplace Transforms 48 Introduction
37
A Few Remarks on the Theory
49
Applications to Differential Equations
68
Derivatives and Integrals of Laplace Transforms
19
Convolutions and Abels Mechanical Problem
22
More about Convolutions The Unit Step and Impulse Functions
27
Appendix A Laplace Appendix B Abel
32
Systems of First Order Equations 54 General Remarks on Systems
3
Linear Systems
5
Homogeneous Linear Systems with Constant Coefficients
10
Nonlinear Systems Volterras PreyPredator Equations
21
Nonlinear Equations 58 Autonomous Systems The Phase Plane and Its Phenomena
26
Types of Critical Points Stability
29
Critical Points and Stability for Linear Systems
79
Stability By Liapunovs Direct Method
86
Simple Critical Points of Nonlinear Systems
90
Nonlinear Mechanics Conservative Systems
90
Introduction A Review of Power Series
26
Series Solutions of First Order Equations
27
Second Order Linear Equations Ordinary Points
28
The Existence and Uniqueness of Solutions
51
Numerical Methods
64
Numerical Tables
80
Answers
95
Index
108
Regular Singular Points 30 Regular Singular Points Continued 31 Gausss Hypergeometric Equation
144
Derechos de autor

Otras ediciones - Ver todas

Differential Equations with Applications and Historical Notes
George F. Simmons
Vista previa limitada - 2016
Differential Equations: With Applications and Historical Notes
George Finlay Simmons
Vista de fragmentos - 1972
Differential Equations with Applications and Historical Notes
George F. Simmons
Sin vista previa disponible - 2023
Ver todas »

Términos y frases comunes

a₁ applied approximation assume asymptotically stable b₁ Bessel functions boundary conditions c₁ calculation calculus of variations called closed path coefficients consider constant continuous converges coordinates corresponding cosine series critical point 0,0 curve derivatives differential equation Dirichlet discussion dx dt dxdt dy dt dy dx eigenfunctions Euler method Euler's equation Example expansion Figure formula Fourier series function f(x Gauss geometry given graph initial conditions integral interval Laplace transforms Laplace's equation Legendre Legendre polynomials linear system linearly independent m₁ mathematicians mathematics nonlinear nx dx obtain order equations orthogonal particle particular solution phase plane physical polynomials positive integer positive number power series proof properties prove regular singular point result Riemann satisfies Section sequence sinx solving surface theorem theory value problem variable Wronskian x₁ x²y y₁ yields zero λη ду

Acerca del autor (2016)

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).

Información bibliográfica

TítuloDifferential Equations with Applications and Historical Notes
Textbooks in Mathematics
AutorGeorge F. Simmons
Edición3, revisada
EditorCRC Press, 2016
ISBN1498702627, 9781498702621
Largo764 páginas
  
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