Ordinary Differential Equations Summary:
By L. Ince
Publisher: Dover Publications
Number Of Pages: 558
Publication Date: 1956-06-01
ISBN-10 / ASIN: 0486603490
ISBN-13 / EAN: 9780486603490
Product Description:
Among the topics covered in this classic treatment are linear differential equations; solution in an infinite form; solution by definite integrals; algebraic theory; Sturmian theory and its later developments; further developments in the theory of boundary problems; existence theorems, equations of first order; nonlinear equations of higher order; linear equations in complex domain; and oscillation theorems in the complex domain.
Excerpt:
IN accordance with the tradition which allows an author to make his preface serve rather as an epilogue, I submit that my aim has been to introduce the student into the field of Ordinary Differential Equations, and thereafter to guide him to this or that standpoint from which he may see the outlines of unexplored territory. Naturally, I have not covered the whole domain of the subject, but have chosen a path which I myself have followed and found interesting. If the reader would pause at any point where I have hurried on, or if he would branch off into other tracks, he may seek guidance in the footnotes. In the earlier stages I ask for little outside knowledge, but for later developments I do assume a growing familiarity with other branches of Analysis. For some time I have felt the need for a treatise on Differential Equations whose scope would embrace not merely that body of theory which may now be regarded as classical, but which would cover, in some aspects at least, the main developments which have taken place in the last quarter of a century. During this period, no comprehensive treatise on the subject has been published in England, and very little work in this particular field has been carried out ; while, on the other hand, both on the Continent and in America investigations of deep interest and fundamental importance have been recorded. The reason for this neglect of an important branch of Analysis is that England has but one school of Pure Mathematics, which implies a high development in certain fields and a comparative neglect of others. To spread the energies of this school over the whole domain of Pure Mathematics would be to scatter and weaken its forces ; consequently its interests, which were at no time particularly devoted to the subject of Differential Equations, have now turned more definitely into other channels, and that subject is denied the cultivation which its importance deserves. The resources of those more fortunate countries, in which several schools of the first rank flourish, are adequate to deal with all branches of Mathematics. For this reason, and because of more favourable traditions, the subject of Differential Equations has not elsewhere met with the neglect which it has suffered in England. In a branch of Mathematics with a long history behind it, the prospective investigator must undergo a severer apprenticeship than in a field more recently opened. This applies in particular to the branch of Analysis which lies before us, a branch in which the average worker cannot be certain of winning an early prize. Nevertheless, the beginner who has taken the pains to acquire a sound knowledge of the broad outlines of the subject will find manifold opportunities for original work in a special branch. For instance, I may draw attention to the need for an intensive study of the groups of functions defined by classes of linear equations which have a number of salient features in common.
Contents:
Ordinary Differential Equations Front Cover
Back Cover
PREFACE
CONTENTS
Table of Contents
Part I. Differential Equations in the Real Domain
Chapter I. Introduction
1-1. Definitions
1-2. Genesis of an Ordinary Differential Equation
1-201. The Differential Equation of a Family of Confocal Conies
1-21. Formation of Partial Differential Equations through the Elimination o! Arbitrary Constants
1-22. A Property of Jacobians
1-23. Formation of a Partial Differential Equation through the Elimination ol an Arbitrary Function
1-232. Eider's Theorem on Homogeneous Functions
1-24. Formation of a Total Differential Equation in Three Variables. The equation
1-3. The Solutions of an Ordinary Differential Equation
1-4. Geometrical Significance of the Solutions of an Ordinary Differential Equation of the First Order.
1-5. Simultaneous Systems of Ordinary Differential Equations,
Chapter II. Elementary Methods of Integration
2-1. Exact Equations of the First Order and o! the First Degree
2-2. The Integrating Factor
2-3. Orthogonal Trajectories
2-4. Equations of the First Order but not of the First Degree
2-5. The Principle of Duality
2-6. Equations of Higher Order than the First
2'7. Simultaneous Systems in Three Variables
2'8, Total Differential Equations
Chapter III. The Existence and Nature of Solutions of Ordinary Differential Equations
3-1. Statement Of the Problem
3-2. The Method of Successive Approximations.
3-3. Extension of the Method of Successive Approximation to a System ofEquations of the First Order
3'4. The Cauchy-Lipschitz Method
3-5. Discussion of the Existence Theorem for an Equation not of the FirstDegree.
3' 6. Singular Solutions
3-7. Discussion of a Special Differential Equation.
Chapter IV. Continuous Transformation-Groups
4-1. Lie's Theory of Differential Equations
4-2. Functions Invariant under a Given Group.
4-3. Extension to n Variables
4-4. Determination of all Equations which admit of a given Group
4-5. The Extended Group
4-6. Integration of a Differential Equation of the First Order in Two Variables
Chapter V. The General Theory of Linear Differential Equations
5-1. Properties of a Linear Differential Operator
5-2. The Wronskian
5-3. The Adjoint Equation
5-4. Solutions common to two Linear Differential Equations
5-5. Permutable Linear Operators
Chapter VI. Linear Equations with Constant Coefficients
6-1. The Linear Operator with Constant Coefficients
6-2. Discussion of the Non-Homogeneous Equation
6-3. The Euler Linear Equation
6-4. Systems of Simultaneous Linear Equations with Constant Goefficients.
6-5. Redaction of a System of Linear Equations to the Equivalent DiagonalSystem.
6-6. Behaviour at Infinity of Solutions of a Linear Differential System withbounded Coefficients.
Chapter VII. The Solution of Linear Differential Equations in an Infinite Form
7-1. Failure of the Elementary Methods
7-2. Solutions relative to an Ordinary Point
7-3. The Point at Infinity as an Irregular Singular Point
7-4. Equations with Periodic Coefficients ; the Mathieu Equation
7-5. A connexion between Differential Equations and Continued Fractions
Chapter VIII. The Solution of Linear Differential Equations by Definite Integrals
8-1. The General Principle
8-2. The Laplace Transformation
8-3. The Nucleus
8-4. The Mellin Transformation.
8-5. Solution by Double Integrals
8-6. Periodic Transformations
Chapter IX. The Algebraic Theory of Linear Differential Systems
9-1. Definition of a Linear Differential System
9-2. Analogy with the Theory of a System of Linear Algebraic Equations.
9-3. Properties of a Bilinear Form
9-4. The self-adjoint Linear Differential System of the Second Order
9-5. Differential Systems which involve a Parameter. The Characteristic Numbers.
9-6. The Effect of Small Variations in the Coefficients of a Linear Differential System
Chapter X. The Sturmian Theory and its Later Developments
10-1. The Purpose ol the Sturmian Theory
10-2. The Separation Theorem
10'3. Sturm's Fundamental Theorem
10-4. The First Comparison Theorem
10-5. Boundary Problems in One Dimension
10-6. Sturm's Oscillation Theorems
10-7. The Orthogonal Property o! Characteristic Functions and its Consequences
10-8. Periodic Boundary Conditions.
10-9. Klein's Oscillation Theorem
Chapter XI. Further Developments in the Theory of Boundary Problems
11-1. Green's Functions in One Dimension
11-2. The Relationship between a Linear Differential System and an Integral Equation.
11-3. Application of the Method of Successive Approximations
11-4 The Asymptotic Development of Characteristic Numbers and Functions
11-5. The Sturm-Liouville Development of an Arbitrary Function
Part II. Differential Equations in the Complex Domain
Chapter XII. Existence Theorems in the Complex Domain
12-1. General Statement
12-2. The Method of Limits
12-3. Analytical Continuation of the Solution ; Singular Points
12-4. Initial Values for which f(z, w) is Infinite
12-5. Fixed and Movable Singular Points.
12-6. Initial Values for which f(z, w) is indeterminate
Chapter XIII. Equations of the First Order But Not of the First Degree
13-1. Specification oi the Equations Considered
13-2. Case
13-2. Case (i)
13-3. Case (ii)
13-4. Case (iii)
13-5. Case (iv).
13-6. The Dependent Variable initially Infinite
13-7. Equations into which z does not enter explicitly.
13-8. Binomial Equations of Degree
Chapter XIV. Non-Linear Equations of Higher Order
14-1. Statement of the Problem
14-2. Application of the Method
14-3. Reduction to Standard Form
14-4. The Painleve Transcendents
Chapter XV. Linear Equations in the Complex Domain
15-1. The a priori Knowledge of the Singular Points.-
15-2. Closed Circuits enclosing Singular Points
15-3. A Necessary Condition for a Regular Singularity.
15-4. Equations of Fuchsian Type
15-5. A Glass of Equations whose general Solution is Uniform
15-6. Equations whose Coefficients are Doubly-Periodic Functions
15'7. Equations with Simply-Periodic Coefficients
15'8. Analogies with the Puchsian Theory.
15-9. Linear Substitutions
Chapter XVI. The Solution of Linear Differential Equations in Series
16-1. The Method ot Frobenius
16-2. The Convergence of the Development
16-3. The Solutions corresponding to a Set of Indices
16-4. Real and Apparent Singularities
16-5. The Peano-Baker Method of Solution
Chapter XVII. Equations with Irregular Singular Points
17-1. The Possible Existence of Regular Solutions
17-2. The Indicia! Equation
17-3. Proof of the general Non-Existence o! Regular Solutions
17-4. The Adjoint Equation
17-5. Normal Solutions
17-6. Hamburger Equations
Chapter XVIII. The Solution of Linear Differential Equations by Methods of Contour Integration
18-1. Extension of the Scope of the Laplace Transformation
18-2. Discussion of the Laplace Transformation in the more general Case.
18-3. Equations of Rank greater than Unity : Indirect Treatment
18-4. Integrals of Jordan and Pochhammer
18-5. The Legendre Function
18-6. The Confluent Hypergepmetric Functions.
18-7. The Bessel Functions
Chapter XIX. Systems of Linear Equations of the First Order
19-1. Equivalent Singular Points
19-2. Reduction to a Canonical System
19-3. Formal Solutions
19-4. Solution of the Standard Canonical System of Bank Unity by Laplace Integrals
19-5. Asymptotic Representations
19-6. Characterisation of the Solutions in the Neighbourhood o! Infinity
19-7. The Generalised Riemann Problem
Chapter XX. Classification of Linear Differential Equations of the Second Order with Rational Coefficients
20-1, The Necessity tor a Systematic Classification
20-2. The Confluence of Singular Points.
20-3. Equations derived from the Equation with four Elementary Singularities
20-4. Constants-in-Excess
20-5. Sequences of Equations with Regular Singularities
20-6. Asymptotic Behaviour of Solutions at an Irregular Singularity.
Chapter XXI. Oscillation Theorems in the Complex Domain
21-1. Statement of the Problem
21-2. The Green's Transform
21-3. Selection of an appropriate Path of Integration
21-4 Zero-Free Intervals on the Real Axis
21-5. Asymptotic Distribution of the Zeros
APPENDIX A HISTORICAL NOTE ON FORMAL METHODS OF INTEGRATION
APPENDIX B NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS
APPENDIX C LIST OF JOURNALS QUOTED IN FOOTNOTES TO THE TEXT
APPENDIX D BIBLIOGRAPHY
INDEX OF AUTHORS
GENERAL INDEX
Summary: Very useful for theoretical physicists
Rating: 5
Readable, has a simple chapter on continuous groups that (implicitly) introduces the notion of global integrability. Discusses and uses Fuch's theorem (classification of singularities of linear ode's, basis for 'guessing' the right form of the series solution in terms of singilarities of coefficients), easy group theoretic discussion of singularities in the complex plane. Stage 2: see Arnol'd's Ordinary Differential Equations for theory, Bender and Orszag for approximation methods.
Summary: An essential reference work for anyone working in ode's
Rating: 5
This classic (originally published in 1926 and still in print!) combines readability with a vast wealth accurately presented material (much of which can still only be found in research papers and certainly can nowhere else be found in a single reference). Most astpects of theory are illustrated by examples.
The main areas covered in the book are existence theorems, transformation group (Lie group) methods of solution, linear systems of equations, boundary eigenvalue problems, nature and methods of solution of regular, singular and nonlinear equation in the complex plane, Green's functions for complex equations.
This is an essential reference for anyone working with ordinary differential equations.
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