Handbook of Ordinary Differential Equations - Kamke E. Handbook of First Order Partial Differential Equations - Kamke E Kamke Handbook of Ordinary Differential Equations

Name: A Handbook of Ordinary Differential Equations.

The "Handbook of Ordinary Differential Equations" by the famous German mathematician Erich Kamke (1890 - 1961) is a unique edition in terms of material coverage and occupies a worthy place in the world reference mathematical literature.
The first edition of the Russian translation of this book appeared in 1951. The past two decades have been a period of rapid development of computational mathematics and computer technology. Modern computing tools allow quickly and with great accuracy to solve various problems that previously seemed too cumbersome. In particular, numerical methods are widely used in problems related to ordinary differential equations. Nevertheless, the possibility of writing down the general solution of one or another differential equation or system in closed form has in many cases significant advantages. Therefore, the extensive reference material that is collected in the third part of E. Kamke's book - about 1650 equations with solutions - remains of great importance even now.

In addition to the indicated reference material, E. Kamke's book contains a presentation (albeit without proofs) of the basic concepts and most important results related to ordinary differential equations. It also covers a number of such issues that are usually not included in textbooks on differential equations (for example, the theory of boundary value problems and eigenvalue problems).
E. Kamke's book contains many facts and results useful in everyday work; it turned out to be valuable and necessary for a wide range of scientists and specialists in applied fields, for engineers and students. Three previous editions of the translation of this handbook into Russian were welcomed by readers and sold out long ago.
The Russian translation was re-checked against the sixth German edition (1959); fixed inaccuracies, errors and typos. All insertions, comments and additions made in the text by the editor and translator are enclosed in square brackets. At the end of the book, under the heading "Supplements", there are abridged translations (performed by N. Kh. Rozov) of those few journal articles supplementing the reference part that the author mentioned in the sixth German edition.

PART ONE
GENERAL SOLUTION METHODS
Chapter I
§ 1. Differential equations resolved with respect to
derivative: y" \u003d f (x, y); basic concepts
1.1. Notation and geometric meaning of the differential
equations
1.2. Existence and uniqueness of a solution
§ 2. Differential equations resolved with respect to
derivative: y" \u003d f (x, y); solution methods
2.1. Polyline method
2.2. Picard-Lindelöf method of successive approximations
2.3. Application of power series
2.4. A more general case of series expansion25
2.5. Expansion in a series in parameter 27
2.6. Connection with partial differential equations27
2.7. Evaluation theorems 28
2.8. Behavior of Solutions for Large Values ​​x 30
§ 3. Differential equations not resolved with respect to 32
derivative: F(y", y, x)=0
3.1. About solutions and methods of solution 32
3.2. Regular and singular linear elements33
§ 4. Solution of particular forms of differential equations of the first 34
order
4.1. Differential equations with separable variables 35
4.2. y"=f(ax+by+c) 35
4.3. Linear differential equations 35.
4.4. Asymptotic behavior of solutions of linear differential equations
4.5. Bernoulli equation y"+f(x)y+g(x)ya=0 38
4.6. Homogeneous differential equations and their reductions38
4.7. Generalized homogeneous equations 40
4.8. Special Riccati equation: y "+ y2 \u003d bxa 40
4.9. General Riccati equation: y"=f(x)y2+g(x)y+h(x)41
4.10. Abel equation of the first kind44
4.11. Abel equation of the second kind47
4.12. Equation in Total Differentials 49
4.13. Integrating factor 49
4.14. F(y",y,x)=0, "integration by differentiation" 50
4.15. (a) y=G(x, y"); (b) x=G(y, y") 50
4.16. (a) G(y ",x)=0; (b) G(y\y)=Q 51
4.17. (a) y"=g(y); (6) x=g(y") 51
4.18. Clairaut's equations 52
4.19. Lagrange-D'Alembert equation 52
4.20. F(x, xy"-y, y")=0. Legendre transformation53
Chapter II. Arbitrary systems of differential equations solved with respect to derivatives
§ 5. Basic concepts54
5.1. Notation and geometric meaning of the system of differential equations
5.2. Existence and uniqueness of a solution 54
5.3. Carathéodory's existence theorem 5 5
5.4. Dependence of the solution on initial conditions and parameters56
5.5. Sustainability Issues57
§ 6. Solution methods 59
6.1. Polyline method59
6.2. Picard-Lindelöf method of successive approximations59
6.3. Application of power series 60
6.4. Connection with partial differential equations 61
6.5. System reduction using a known relation between solutions
6.6. System reduction by differentiation and elimination 62
6.7. Evaluation theorems 62
§ 7. Autonomous systems 63
7.1. Definition and geometric meaning of an autonomous system 64
7.2. On the behavior of integral curves in a neighborhood of a singular point in the case n = 2
7.3. Criteria for determining the type of singular point 66
Chapter III.
§ 8. Arbitrary linear systems70
8.1. General remarks70
8.2. Existence and uniqueness theorems. Solution methods70
8.3. Reduction of an inhomogeneous system to a homogeneous one71
8.4. Evaluation theorems 71
§ 9. Homogeneous linear systems72
9.1. Solution properties. Fundamental decision systems 72
9.2. Existence theorems and solution methods 74
9.3. System Reduction to a System With Fewer Equations75
9.4. Conjugate system of differential equations76
9.5. Self-adjoint systems of differential equations, 76
9.6. Conjugate systems of differential forms; Lagrange identity, Green's formula
9.7. Fundamental solutions78
§ten. Homogeneous linear systems with singular points 79
10.1. Classification of singular points 79
10.2. Weak singular points80
10.3. Strongly singular points 82
§eleven. Behavior of Solutions for Large Values ​​of x 83
§12. Linear systems depending on a parameter84
§13. Linear systems with constant coefficients 86
13.1. Homogeneous systems 83
13.2. More general systems 87
Chapter IV. Arbitrary differential equations of the nth order
§ 14. Equations resolved with respect to the highest derivative: 89
yin)=f(x,y,y\...,y(n-\))
§fifteen. Equations not resolved with respect to the highest derivative:90
F(x,y,y\...,y(n))=0
15.1. Equations in Total Differentials90
15.2. Generalized homogeneous equations 90
15.3. Equations not explicitly containing x or y 91
Chapter V Linear differential equations of the nth order,
§16. Arbitrary linear differential equations of nth order92
16.1. General remarks92
16.2. Existence and uniqueness theorems. Solution methods92
16.3. Elimination of the (n-1)th order derivative94
16.4. Reduction of an inhomogeneous differential equation to a homogeneous one
16.5. Behavior of Solutions for Large Values ​​of x94
§17. Homogeneous linear differential equations of nth order 95
17.1. Properties of Solutions and Existence Theorems 95
17.2. Lowering the Order of a Differential Equation96
17.3. 0 zero solutions 97
17.4. Fundamental solutions 97
17.5. Conjugate, self-adjoint, and anti-self-adjoint differential forms
17.6. Lagrange identity; Dirichlet and Green's formulas 99
17.7. On solutions of adjoint equations and equations in total differentials
§eighteen. Homogeneous linear differential equations with singular101
dots
18.1. Classification of singular points 101
18.2. The case when the point x=E is regular or weakly singular104
18.3. The case when the point x=inf is regular or weakly singular108
18.4. The case when the point x = % is strongly singular 107
18.5. The case when the point x=inf is strongly singular 108
18.6. Differential Equations with Polynomial Coefficients
18.7. Differential Equations with Periodic Coefficients
18.8. Differential Equations with Doubly Periodic Coefficients
18.9. The case of a real variable112
§19. Solving linear differential equations using 113
definite integrals
19.1. General principle 113
19.2. Laplace transform 116
19.3 Special Laplace Transform 119
19.4. Mellin Transform 120
19.5. Euler transform 121
19.6. Solution using double integrals 123
§ 20. Behavior of solutions for large values ​​of x 124
20.1. Polynomial Coefficients124
20.2. More general coefficients 125
20.3. Continuous odds 125
20.4. Oscillation theorems126
§21. Linear differential equations of nth order depending on127
parameter
§ 22. Some special types of linear differentials129
nth order equations
22.1. Homogeneous differential equations with constant coefficients
22.2. Inhomogeneous differential equations with constants130
22.3. Euler Equations 132
22.4. Laplace equation132
22.5. Equations with polynomial coefficients133
22.6. Pochhammer equation134
Chapter VI. Second order differential equations
§ 23. Nonlinear differential equations of the second order 139
23.1. Methods for solving particular types of nonlinear equations 139
23.2. Some additional remarks140
23.3. Limit value theorems 141
23.4. Oscillation Theorem 142
§ 24. Arbitrary linear differential equations of the second 142
order
24.1. General remarks142
24.2. Some methods of solving 143
24.3. Evaluation theorems 144
§ 25. Homogeneous second-order linear differential equations 145
25.1. Reduction of second-order linear differential equations
25.2. Further Remarks on the Reduction of Second-Order Linear Equations
25.3. Expanding the Solution into a Continued Fraction 149
25.4. General remarks about solution zeros150
25.5. Zeros of Solutions on a Finite Interval151
25.6. Behavior of solutions for x->inf 153
25.7. Second Order Linear Differential Equations with Singular Points
25.8. Approximate solutions. Asymptotic solutions real variable
25.9. Asymptotic solutions; complex variable161
25.10. WBC method 162
Chapter VII. Linear differential equations of the third and fourth
orders
§ 26. Linear differential equations of the third order163
§ 27. Linear differential equations of the fourth order 164
Chapter VIII. Approximate methods for integrating differential
equations
§ 28. Approximate integration of differential equations 165
first order
28.1. The method of broken lines165.
28.2. Additional Half Step Method 166
28.3. Runge-Hein-Kutta method 167
28.4. Combining interpolation and successive approximations168
28.5. Adams Method 170
28.6. Additions to the Adams method 172
§ 29. Approximate integration of differential equations 174
higher orders
29.1. Approximate Integration Methods for Systems of First-Order Differential Equations
29.2. The broken line method for second-order differential equations 176
29.3. Runge-Kutta Method for Second Order Differential Equations
29.4. Adams - Shtormer method for the equation y "=f (x, y, y) 177
29.5. Adams - Shtormer method for the equation y "=f (x, y) 178
29.6. Bless's method for the equation y"=f(x,y,y) 179

PART TWO
Boundary value and eigenvalue problems
Chapter I Boundary Value Problems and Eigenvalue Problems for Linear
differential equations of nth order
§ 1. General theory of boundary value problems182
1.1. Notation and Preliminaries 182
1.2. Conditions for the solvability of a boundary value problem184
1.3. Conjugate boundary value problem 185
1.4. Self-adjoint boundary value problems 187
1.5. Green's function 188
1.6. Solving an inhomogeneous boundary value problem using the Green's function 190
1.7. Generalized Green's function 190
§ 2. Boundary Value Problems and Eigenvalue Problems for the Equation 193
£SHU(Y)+YX)Y = 1(X)
2.1. Eigenvalues ​​and eigenfunctions; characteristic determinant A(X)
2.2. Adjoint eigenvalue problem and Green's resolvent; complete biorthogonal system
2.3. Normalized boundary conditions; regular eigenvalue problems
2.4. Eigenvalues ​​for regular and irregular eigenvalue problems
2.5. Expansion of a Given Function in Eigenfunctions of Regular and Irregular Eigenvalue Problems
2.6. Self-adjoint normal eigenvalue problems 200
2.7. On Integral Equations of Fredholm Type 204
2.8. Relationship between boundary value problems and integral equations of the Fredholm type
2.9. Relationship between eigenvalue problems and integral equations of Fredholm type
2.10. On Integral Equations of Volterra Type211
2.11. Relationship between boundary value problems and integral equations of Volterra type
2.12. Relationship between eigenvalue problems and integral equations of Volterra type
2.13. Relationship between eigenvalue problems and the calculus of variations
2.14. Application to expansion in terms of eigenfunctions218
2.15. Additional remarks219
§ 3. Approximate methods for solving problems on eigenvalues ​​u222-
boundary value problems
3.1. Approximate Galerkin-Ritz method222
3.2. Approximate Grammel method224
3.3. Solving an inhomogeneous boundary value problem using the Galerkin-Ritz method
3.4. Method of successive approximations 226
3.5. Approximate solution of boundary value problems and eigenvalue problems by the method of finite differences
3.6. Perturbation method 230
3.7. Eigenvalue estimates 233
3.8. Overview of ways to calculate eigenvalues ​​and 236 eigenfunctions
§ 4. Self-adjoint eigenvalue problems for an equation238
F(y)=W(y)
4.1. Problem Statement 238
4.2. General preliminary remarks 239
4.3. Normal eigenvalue problems 240
4.4. Positive definite eigenvalue problems 241
4.5. Eigenfunction expansion 244
§ 5. Boundary and additional conditions of a more general form 247
Chapter II. Boundary Value Problems and Eigenvalue Problems for Systems
linear differential equations
§ 6. Boundary value problems and eigenvalue problems for systems 249
linear differential equations
6.1. Notation and solvability conditions 249
6.2. Conjugate boundary value problem 250
6.3. Green Matrix252
6.4. Eigenvalue Problems 252-
6.5. Self-adjoint eigenvalue problems 253
Chapter III. Boundary Value Problems and Eigenvalue Problems for Equations
lower orders
§ 7. Problems of the first order256
7.1. Linear Problems 256
7.2. Nonlinear Problems 257
§ 8. Linear boundary value problems of the second order257
8.1. General remarks 257
8.2. Green's function 258
8.3. Estimates for solutions of boundary value problems of the first kind259
8.4. Boundary conditions for |х|->inf259
8.5. Finding Periodic Solutions 260
8.6. One boundary value problem related to the study of fluid flow 260
§ 9. Linear eigenvalue problems of the second order 261
9.1. General remarks 261
9.2 Self-adjoint eigenvalue problems 263
9.3. y"=F(x,)Cjz, z"=-G(x,h)y and the boundary conditions are self-adjoint266
9.4. Eigenvalue problems and the variational principle269
9.5. On the practical calculation of eigenvalues ​​and eigenfunctions
9.6. Eigenvalue problems, not necessarily self-adjoint271
9.7. Additional conditions of a more general form273
9.8. Eigenvalue Problems Containing Multiple Parameters
9.9. Differential Equations with Singularities at Boundary Points 276
9.10. Eigenvalue problems on an infinite interval 277
§ten. Nonlinear Boundary Value Problems and Eigenvalue Problems 278
second order
10.1. Boundary Value Problems for a Finite Interval 278
10.2. Boundary value problems for a semibounded interval 281
10.3. Eigenvalue problems282
§eleven. Boundary Value Problems and Eigenvalue Problems of the Third
eighth order
11.1. Linear eigenvalue problems of the third order283
11.2. Linear eigenvalue problems of the fourth order 284
11.3. Linear problems for a system of two second-order differential equations
11.4. Nonlinear Boundary Value Problems of the Fourth Order 287
11.5. Higher Order Eigenvalue Problems288

PART THREE
SEPARATE DIFFERENTIAL EQUATIONS
Preliminary remarks 290
Chapter I First order differential equations
1-367. Differential, equations of the first degree with respect to U 294
368-517. Second degree differential equations with respect to 334
518-544. Third degree differential equations with respect to 354
545-576. Differential Equations of a More General Form358
Chapter II. Second Order Linear Differential Equations
1-90. ay" + ...363
91-145. (ax + yuy " + ... 385
146-221.x2 y" + ... 396
222-250. (x2 ± a2) y "+ ... 410
251-303. (ax2 + bx + c) y" + ... 419
304-341. (ax3 +...)y" + ...435
342-396. (ax4 +...)y" + ...442
397-410. (ah "+ ...) y" + ... 449
411-445. Other Differential Equations 454
Chapter III. Linear differential equations of the third order
Chapter IV. Linear differential equations of the fourth order
Chapter V Linear differential equations of the fifth and higher
orders
Chapter VI. Nonlinear Second Order Differential Equations
1-72. ay"=F(x,y,y)485
73-103./(x);y"=F(x,;y,;y") 497
104- 187. / (x) xy "CR (x,; y,; y") 503
188-225. f(x,y)y"=F(x,y,y)) 514
226-249. Other Differential Equations 520
Chapter VII. Nonlinear differential equations of the third and more
high orders
Chapter VIII. Systems of linear differential equations
Preliminary remarks 530
1-18. Systems of two differential equations of the first order с530
constant coefficients 19-25.
Systems of two differential equations of the first order с534
variable coefficients
26-43. Systems of two differential equations of order above535
first
44-57. Systems of more than two differential equations538
Chapter IX. Systems of nonlinear differential equations
1-17. Systems of two differential equations541
18-29. Systems of more than two differential equations 544
ADDITIONS
On the solution of linear homogeneous equations of the second order (I. Zbornik) 547
Additions to the book by E. Kamke (D. Mitrinovich) 556
A new way to classify linear differential equations and 568
constructing their general solution using recursive formulas
(I. Zbornik)
Index 571

Kamke E. A Handbook of First-Order Partial Differential Equations: A Handbook. Edited by N.X. Rozova - M.: "Nauka", 1966. - 258 p.
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However, very recently, interest in partial differential equations of the first order has again strongly increased. Two factors contributed to this. First of all, it turned out that the so-called generalized solutions of first-order quasilinear equations are of exceptional interest for applications (for example, in the theory of shock waves in gas dynamics, etc.). In addition, the theory of systems of partial differential equations has stepped far ahead. Nevertheless, to date, there is no monograph in Russian that would collect and present all the facts accumulated in the theory of partial differential equations of the first order, except for the well-known book by N. M. Gyun-

FOREWORD TO THE RUSSIAN EDITION

tera, which has long become a bibliographic rarity. This book fills this gap to some extent.

The name of Professor E. Kamke of the University of Tübingen is familiar to Soviet mathematicians. He owns a large number of works on differential equations and some other branches of mathematics, as well as several books of an educational nature. In particular, his monograph "The Lebesgue-Stieltjes Integral" was translated into Russian and published in 1959. Three editions in Russian in 1951, 1961, 1965 were issued by the "Handbook of Ordinary Differential Equations", which is a translation of the first volume of "Gewohnliche Differenlialglelchungen" of E. Kamke's book "Differentialgleichungen (Losungsmethoden und L6sungen)".

"Handbook of First-Order Partial Differential Equations" is a translation of the second volume of the same book. There are collected about 500 equations with solutions. In addition to this material, this handbook contains a concise (without proof) presentation of a number of theoretical issues, including those that are not included in the usual courses of differential equations, such as existence theorems, uniqueness, etc.

In preparing the Russian edition, the extensive bibliography available in the book was revised. References to old and inaccessible foreign textbooks were replaced, if possible, with references to domestic and translated literature. All noted inaccuracies, errors and typos have been corrected. All inserts, comments and additions made to the book during editing are enclosed in square brackets.

This handbook, created in the early forties (and since then repeatedly reprinted in the GDR without any changes), undoubtedly no longer fully reflects the achievements that are now available in the theory of partial differential equations of the first order. So, the theory of generalized solutions of quasilinear equations, developed in the well-known works of I. M. Gelfand, O. A. Oleinik and others, did not find any reflection in the reference book. Not covered in the handbook and the theory of Pfaff equations. However, it seems that even in this form the book will undoubtedly prove to be a useful guide to the classical theory of partial differential equations of the first order.

The summary of equations given in the book, the solutions of which can be written down in the final form, is very interesting and useful, but, of course, is not exhaustive. It was compiled by the author on the basis of works that appeared before the beginning of the forties.

SOME NOTATIONS

x, y; hee xp; yi .... yn - independent variables, r- (x (, xn) a, b, c; A, B, C - constants, constant coefficients, @, @ (x, y), @ (r) - open region, region on the plane (x, y), in the space of variables xt,...,xn [usually the region of continuity of coefficients and solutions. - Note ed.], g - subdomain @, F, f - general function,

fi - arbitrary function, r; r(x, y); z - ty(x....., xn) - desired function, solution,

Dg _ dg _ dg _ dg

p~~dx "q~~dy~" Pv~lx^" qv~~dy~^"

x, |A, k, n - summation indices,

\n)~n! (n - t)! "

/g„...zln\

det | zkv\ - determinant of the matrix I.....I.

\gsh - gpp I

ACCEPTED ABBREVIATIONS IN BIBLIOGRAPHICAL INSTRUCTIONS

Günther - N. M. Günter, Integration of first-order partial differential equations, GTTI, 1934.

Kamke - E. Kamke, Handbook of Ordinary Differential Equations, Nauka, 1964.

Courant - R. Courant, Partial Differential Equations, Mir, 1964.

Petrovsky - I. G. Petrovsky, Lectures on the theory of ordinary differential equations, "Nauka", 1964.

Stepanov - V. V. Stepanov, Course of differential equations, Fizmat-giz, 1959.

Kamke, DQlen-E. Kamke, Differentialgleichungen reeller Funktionen, Leipzig, 1944.

The abbreviations of the names of periodicals correspond to the generally accepted ones and therefore are omitted in the translation; see, however, K a m to e. - Approx. ed.]

PART ONE

GENERAL SOLUTION METHODS

[The following literature is devoted to the issues considered in the first part:

Per. with him. - 4th ed., Rev. - M.: Science: Ch. ed. physics and mathematics lit., 1971. - 576s.

FROM THE PREFACE TO THE FOURTH EDITION

The "Handbook of Ordinary Differential Equations" by the famous German mathematician Erich Kamke (1890-1961) is a unique edition in terms of material coverage and occupies a worthy place in the world reference mathematical literature.

The first edition of the Russian translation of this book appeared in 1951. The past two decades have been a period of rapid development of computational mathematics and computer technology. Modern computing tools allow quickly and with great accuracy to solve various problems that previously seemed too cumbersome. In particular, numerical methods are widely used in problems related to ordinary differential equations. Nevertheless, the possibility of writing down the general solution of one or another differential equation or system in closed form has in many cases significant advantages. Therefore, the extensive reference material that is collected in the third part of E. Kamke's book - about 1650 equations with solutions - remains of great importance even now.

In addition to the indicated reference material, E. Kamke's book contains a presentation (albeit without proofs) of the basic concepts and most important results related to ordinary differential equations. It also covers a number of such issues that are usually not included in textbooks on differential equations (for example, the theory of boundary value problems and eigenvalue problems).

E. Kamke's book contains many facts and results useful in everyday work; it turned out to be valuable and necessary for a wide range of scientists and specialists in applied fields, for engineers and students. Three previous editions of the translation of this handbook into Russian were welcomed by readers and sold out long ago.

  • Table of contents
  • Preface to the fourth edition 11
  • Some designations 13
  • Accepted abbreviations in bibliographic indications 13
  • PART ONE
  • GENERAL SOLUTION METHODS Chapter I. First Order Differential Equations
  • § 1. Differential equations solved with respect to 19
  • derivative: at" =f(x,y); basic concepts
  • 1.1. Notation and geometric meaning of the differential 19
  • equations
  • 1.2. Existence and uniqueness of a solution 20
  • § 2. Differential equations solved with respect to 21
  • derivative: at" =f(x,y); solution methods
  • 2.1. Polyline method 21
  • 2.2. Picard-Lindelöf method of successive approximations 23
  • 2.3. Application of power series 24
  • 2.4. A more general case of series expansion 25
  • 2.5. Expansion in a series in parameter 27
  • 2.6. Connection with partial differential equations 27
  • 2.7. Evaluation theorems 28
  • 2.8. Behavior of Solutions for Large Values X 30
  • § 3. Differential equations not solved with respect to 32
  • derivative: F(y", y, x)=0
  • 3.1. About solutions and methods of solution 32
  • 3.2. Regular and singular linear elements 33
  • § 4. Solution of particular forms of differential equations of the first 34
  • order
  • 4.1. Differential equations with separable variables 35
  • 4.2. y"=f(ax+by+c) 35
  • 4.3. Linear differential equations 35.
  • 4.4. Asymptotic behavior of solutions
  • 4.5. Bernoulli equation y"+f(x)y+g(x)y a =0 38
  • 4.6. Homogeneous differential equations and their reductions 38
  • 4.7. Generalized homogeneous equations 40
  • 4.8. Special Riccati Equation: y "+ ay 2 \u003d bx a 40
  • 4.9. General Riccati equation: y"=f(x)y 2 +g(x)y+h(x) 41
  • 4.10. Abel equation of the first kind 44
  • 4.11. Abel equation of the second kind 47
  • 4.12. Equation in Total Differentials 49
  • 4.13. Integrating factor 49
  • 4.14. F(y",y,x)=0, "integration by differentiation" 50
  • 4.15. (a) y=G(x, y"); (b) x=G(y, y") 50 4.16. (a) G(y ",x)=0; (b) G(y y)=Q 51
  • 4L7. (a) y"=g(y); (6) x=g(y") 51
  • 4.18. Clairaut's equations 52
  • 4.19. Lagrange-D'Alembert equation 52
  • 4.20. F(x, xy"-y, y")=0. Legendre transformation 53 Chapter II. Arbitrary systems of differential equations,
  • permitted relative to derivatives
  • § 5. Basic concepts 54
  • 5.1. Notation and geometric meaning of the system of differential equations
  • 5.2. Existence and uniqueness of a solution 54
  • 5.3. Carathéodory's existence theorem 5 5
  • 5.4. Dependence of the Solution on the Initial Conditions and on the Parameters 56
  • 5.5. Sustainability Issues 57
  • § 6. Solution methods 59
  • 6.1. Polyline method 59
  • 6.2. Picard-Lindelöf method of successive approximations 59
  • 6.3. Application of power series 60
  • 6.4. Connection with partial differential equations 61
  • 6.5. System reduction using a known relation between solutions
  • 6.6. System reduction by differentiation and elimination 62
  • 6.7. Evaluation theorems 62
  • § 7. Autonomous systems 63
  • 7.1. Definition and geometric meaning of an autonomous system 64
  • 7.2. On the behavior of integral curves in a neighborhood of a singular point in the case n = 2
  • 7.3. Criteria for determining the type of singular point 66
  • Chapter III. Systems of linear differential equations
  • § 8. Arbitrary linear systems 70
  • 8.1. General remarks 70
  • 8.2. Existence and uniqueness theorems. Solution methods 70
  • 8.3. Reduction of an inhomogeneous system to a homogeneous one 71
  • 8.4. Evaluation theorems 71
  • § 9. Homogeneous linear systems 72
  • 9.1. Solution properties. Fundamental decision systems 72
  • 9.2. Existence theorems and solution methods 74
  • 9.3. Reduction of the system to a system With a smaller number of equations 75
  • 9.4. Conjugate system of differential equations 76
  • 9.5. Self-adjoint systems of differential equations, 76
  • 9.6. Conjugate systems of differential forms; Lagrange identity, Green's formula
  • 9.7. Fundamental solutions 78
  • §ten. Homogeneous linear systems with singular points 79
  • 10.1. Classification of singular points 79
  • 10.2. Weak singular points 80
  • 10.3. Strongly singular points 82 §11. Behavior of Solutions for Large Values X 83
  • §12. Linear systems dependent on parameter 84
  • §13. Linear systems with constant coefficients 86
  • 13.1. Homogeneous systems 83
  • 13.2. More general systems 87 Chapter IV. Arbitrary differential equations nth order
  • § 14. Equations resolved with respect to the highest derivative: 89
  • yin)=f(x,y,y...,y(n-) )
  • §fifteen. Equations not resolved with respect to the highest derivative: 90
  • F(x,y,y...,y(n))=0
  • 15.1. Equations in Total Differentials 90
  • 15.2. Generalized homogeneous equations 90
  • 15.3. Equations not explicitly containing x or at 91 Chapter V. Linear differential equations nth order,
  • §16. Arbitrary linear differential equations nth order 92
  • 16.1. General remarks 92
  • 16.2. Existence and uniqueness theorems. Solution methods 92
  • 16.3. Elimination of the derivative (n-1)th order 94
  • 16.4. Reduction of an inhomogeneous differential equation to a homogeneous one
  • 16.5. Behavior of Solutions for Large Values X 94
  • §17. Homogeneous linear differential equations nth order 95
  • 17.1. Properties of Solutions and Existence Theorems 95
  • 17.2. Reducing the Order of a Differential Equation 96
  • 17.3. 0 zero solutions 97
  • 17.4. Fundamental solutions 97
  • 17.5. Conjugate, self-adjoint, and anti-self-adjoint differential forms
  • 17.6. Lagrange identity; Dirichlet and Green's formulas 99
  • 17.7. On solutions of adjoint equations and equations in total differentials
  • §eighteen. Homogeneous linear differential equations with singular 101
  • dots
  • 18.1. Classification of singular points 101
  • 18.2. The case when the point x=E, regular or weakly singular 104
  • 18.3. The case when the point x=inf is regular or weakly singular 108
  • 18.4. The case when the point x=% strongly special 107
  • 18.5. The case when the point x=inf is strongly singular 108
  • 18.6. Differential Equations with Polynomial Coefficients
  • 18.7. Differential Equations with Periodic Coefficients
  • 18.8. Differential Equations with Doubly Periodic Coefficients
  • 18.9. Case of a real variable 112
  • §19. Solving linear differential equations using 113
  • definite integrals 19.1. General principle 113
  • 19.2. Laplace transform 116
  • 19.3 Special Laplace Transform 119
  • 19.4. Mellin Transform 120
  • 19.5. Euler transform 121
  • 19.6. Solution using double integrals 123
  • § 20. Behavior of solutions for large values X 124
  • 20.1. Polynomial Coefficients 124
  • 20.2. More general coefficients 125
  • 20.3. Continuous odds 125
  • 20.4. Oscillation theorems 126
  • §21. Linear differential equations n-th order depending on 127
  • parameter
  • § 22. Some special types of linear differentials 129
  • equations n-th order
  • 22.1. Homogeneous differential equations with constant coefficients
  • 22.2. Inhomogeneous differential equations with constants 130
  • 22.3. Euler Equations 132
  • 22.4. Laplace Equation 132
  • 22.5. Equations with polynomial coefficients 133
  • 22.6. Pochhammer Equation 134
  • Chapter VI. Second order differential equations
  • § 23. Nonlinear differential equations of the second order 139
  • 23.1. Methods for solving particular types of nonlinear equations 139
  • 23.2. Some additional remarks 140
  • 23.3. Limit value theorems 141
  • 23.4. Oscillation Theorem 142
  • § 24. Arbitrary linear differential equations of the second 142
  • order
  • 24.1. General remarks 142
  • 24.2. Some methods of solving 143
  • 24.3. Evaluation theorems 144
  • § 25. Homogeneous second-order linear differential equations 145
  • 25.1. Reduction of second-order linear differential equations
  • 25.2. Further Remarks on the Reduction of Second-Order Linear Equations
  • 25.3. Expanding the Solution into a Continued Fraction 149
  • 25.4. General remarks about solution zeros 150
  • 25.5. Zeros of Solutions on a Finite Interval 151
  • 25.6. The behavior of solutions for x->inf 153
  • 25.7. Second Order Linear Differential Equations with Singular Points
  • 25.8. Approximate solutions. Asymptotic solutions real variable
  • 25.9. Asymptotic solutions; complex variable 161 25.10. WBC method 162 Chapter VII. Linear differential equations of the third and fourth
  • orders
  • § 26. Linear differential equations of the third order 163
  • § 27. Linear differential equations of the fourth order 164 Chapter VIII. Approximate methods for integrating differential
  • equations
  • § 28. Approximate integration of differential equations 165
  • first order
  • 28.1. The method of broken lines 165.
  • 28.2. Additional Half Step Method 166
  • 28.3. Runge-Hein-Kutta method 167
  • 28.4. Combining interpolation and successive approximations 168
  • 28.5. Adams Method 170
  • 28.6. Additions to the Adams method 172
  • § 29. Approximate integration of differential equations 174
  • higher orders
  • 29.1. Approximate Integration Methods for Systems of First-Order Differential Equations
  • 29.2. The broken line method for second-order differential equations 176
  • 29.3. Runge-Kutta Method for Second Order Differential Equations
  • 29.4. Adams - Shtormer method for the equation y"=f(x,y,y) 177
  • 29.5. Adams - Shtormer method for the equation y"=f(x,y) 178
  • 29.6. Bless's method for equation y"=f(x,y,y) 179
  • PART TWO
  • Boundary Value Problems and Eigenvalue Problems Chapter I. Boundary Value Problems and Eigenvalue Problems for Linear
  • differential equations n-th order
  • § 1. General theory of boundary value problems 182
  • 1.1. Notation and Preliminaries 182
  • 1.2. Conditions for the solvability of a boundary value problem 184
  • 1.3. Conjugate boundary value problem 185
  • 1.4. Self-adjoint boundary value problems 187
  • 1.5. Green's function 188
  • 1.6. Solving an inhomogeneous boundary value problem using the Green's function 190
  • 1.7. Generalized Green's function 190
  • § 2. Boundary Value Problems and Eigenvalue Problems for the Equation 193
  • £w(y) + xx)y = 1(x)
  • 2.1. Eigenvalues ​​and eigenfunctions; characteristic determinant OH)
  • 2.2. Adjoint eigenvalue problem and Green's resolvent; complete biorthogonal system
  • 2.3. Normalized boundary conditions; regular eigenvalue problems 2.4. Eigenvalues ​​for regular and irregular eigenvalue problems
  • 2.5. Expansion of a Given Function in Eigenfunctions of Regular and Irregular Eigenvalue Problems
  • 2.6. Self-adjoint normal eigenvalue problems 200
  • 2.7. On Integral Equations of Fredholm Type 204
  • 2.8. Relationship between boundary value problems and integral equations of the Fredholm type
  • 2.9. Relationship between eigenvalue problems and integral equations of Fredholm type
  • 2.10. On integral equations of Volterra type 211
  • 2.11. Relationship between boundary value problems and integral equations of Volterra type
  • 2.12. Relationship between eigenvalue problems and integral equations of Volterra type
  • 2.13. Relationship between eigenvalue problems and the calculus of variations
  • 2.14. Application to eigenfunction expansion 218
  • 2.15. Additional remarks 219
  • § 3. Approximate methods for solving problems about eigenvalues ​​and 222-
  • boundary value problems
  • 3.1. Approximate Galerkin-Ritz method 222
  • 3.2. Approximate Grammel method 224
  • 3.3. Solving an inhomogeneous boundary value problem using the Galerkin-Ritz method
  • 3.4. Method of successive approximations 226
  • 3.5. Approximate solution of boundary value problems and eigenvalue problems by the method of finite differences
  • 3.6. Perturbation method 230
  • 3.7. Eigenvalue estimates 233
  • 3.8. Overview of ways to calculate eigenvalues ​​and 236 eigenfunctions
  • § 4. Self-adjoint eigenvalue problems for the equation 238
  • F(y)=W(y)
  • 4.1. Problem Statement 238
  • 4.2. General preliminary remarks 239
  • 4.3. Normal eigenvalue problems 240
  • 4.4. Positive definite eigenvalue problems 241
  • 4.5. Eigenfunction expansion 244
  • § 5. Boundary and additional conditions of a more general form 247 Chapter II. Boundary Value Problems and Eigenvalue Problems for Systems
  • linear differential equations
  • § 6. Boundary value problems and eigenvalue problems for systems 249
  • linear differential equations
  • 6.1. Notation and solvability conditions 249
  • 6.2. Conjugate boundary value problem 250
  • 6.3. Green's matrix 252 6.4. Eigenvalue Problems 252-
  • 6.5. Self-adjoint eigenvalue problems 253 Chapter III. Boundary Value Problems and Eigenvalue Problems for Equations
  • lower orders
  • § 7. Problems of the first order 256
  • 7.1. Linear Problems 256
  • 7.2. Nonlinear Problems 257
  • § 8. Linear boundary value problems of the second order 257
  • 8.1. General remarks 257
  • 8.2. Green's function 258
  • 8.3. Estimates for solutions of boundary value problems of the first kind 259
  • 8.4. Boundary conditions for |х|->inf 259
  • 8.5. Finding Periodic Solutions 260
  • 8.6. One boundary value problem related to the study of fluid flow 260
  • § 9. Linear eigenvalue problems of the second order 261
  • 9.1. General remarks 261
  • 9.2 Self-adjoint eigenvalue problems 263
  • 9.3. y"=F(x,)Cjz, z"=-G(x,h)y and the boundary conditions are self-adjoint 266
  • 9.4. Eigenvalue problems and the variational principle 269
  • 9.5. On the practical calculation of eigenvalues ​​and eigenfunctions
  • 9.6. Eigenvalue problems, not necessarily self-adjoint 271
  • 9.7. Additional conditions of a more general form 273
  • 9.8. Eigenvalue Problems Containing Multiple Parameters
  • 9.9. Differential Equations with Singularities at Boundary Points 276
  • 9.10. Eigenvalue problems on an infinite interval 277
  • §ten. Nonlinear Boundary Value Problems and Eigenvalue Problems 278
  • second order
  • 10.1. Boundary Value Problems for a Finite Interval 278
  • 10.2. Boundary value problems for a semibounded interval 281
  • 10.3. Eigenvalue problems 282
  • §eleven. Boundary Value Problems and Eigenvalue Problems of the Third
  • eighth order
  • 11.1. Linear eigenvalue problems of the third order 283
  • 11.2. Linear eigenvalue problems of the fourth order 284
  • 11.3. Linear problems for a system of two second-order differential equations
  • 11.4. Nonlinear Boundary Value Problems of the Fourth Order 287
  • 11.5. Higher order eigenvalue problems 288
  • PART THREE
  • SEPARATE DIFFERENTIAL EQUATIONS
  • Preliminary remarks 290 Chapter I. First-order differential equations
  • 1-367. Differential, first-degree equations with respect to U 294
  • 368-517. Second degree differential equations with respect to 334 518-544. Third degree differential equations with respect to 354
  • 545-576. Differential equations of a more general form 358Chapter II. Second Order Linear Differential Equations
  • 1-90. ay" + ... 363
  • 91-145. (ax + yuy " + ... 385
  • 146-221.x2 y" + ... 396
  • 222-250. (x 2 ± a 2) y "+ ... 410
  • 251-303. (ah 2 + bx + c) y " + ... 419
  • 304-341. (ah 3 +...)y" + ... 435
  • 342-396. (ah 4 +...)y" + ... 442
  • 397-410. (Oh" +...)y" + ... 449
  • 411-445. Other Differential Equations 454
  • G lava III. Third Order Linear Differential Equations Chapter IV. Fourth Order Linear Differential Equations Chapter V. Fifth and Higher Linear Differential Equations
  • Orders Chapter VI. Nonlinear Second Order Differential Equations
  • 1-72. ay"=F(x,y,y) 485
  • 73-103./(x);y"=F(x,;y,;y") 497
  • 104- 187. / (x) xy "CR (x,; y,; y") 503
  • 188-225. f(x,y)y"=F(x,y,y)) 514
  • 226-249. Other differential equations 520Chapter VII. Nonlinear differential equations of the third and more
  • High OrdersChapter VIII. Systems of linear differential equations
  • Preliminary remarks 530
  • 1-18. Systems of two differential equations of the first order with 530
  • constant coefficients 19-25.
  • Systems of two differential equations of the first order with 534
  • variable coefficients
  • 26-43. Systems of two differential equations of order above 535
  • first
  • 44-57. Systems of more than two differential equations 538Chapter IX. Systems of nonlinear differential equations
  • 1-17. Systems of two differential equations 541
  • 18-29. Systems of more than two differential equations 544
  • ADDITIONS
  • On the solution of linear homogeneous equations of the second order (I. Zbornik) 547
  • Additions to the book by E. Kamke (D. Mitrinovich) 556
  • A new way to classify linear differential equations and 568
  • constructing their general solution using recursive formulas
  • (I. Zbornik)
  • Index 571

Preface to the fourth edition
Some designations
Accepted abbreviations in bibliographic indications
PART ONE
GENERAL SOLUTION METHODS
§ 1. Differential equations resolved with respect to the derivative: (formula) basic concepts
1.1. Notation and geometric meaning of the differential equation
1.2. Existence and uniqueness of a solution
§ 2. Differential equations resolved with respect to the derivative: (formula); solution methods
2.1. Polyline method
2.2. Picard-Lindelöf method of successive approximations
2.3. Application of power series
2.4. A more general case of series expansion
2.5. Parameter series expansion
2.6. Relationship with partial differential equations
2.7. Estimate theorems
2.8. Behavior of solutions for large values ​​(?)
§ 3. Differential equations that are not resolved with respect to the derivative: (formula)
3.1. About solutions and solution methods
3.2. Regular and special line elements
§ 4. Solution of Particular Types of First-Order Differential Equations
4.1. Differential equations with separable variables
4.2. (formula)
4.3. Linear differential equations
4.4. Asymptotic behavior of solutions of linear differential equations
4.5. Bednulli equation (formula)
4.6. Homogeneous differential equations and their reductions
4.7. Generalized homogeneous equations
4.8. Special Riccati equation: (formula)
4.9. General Riccati equation: (formula)
4.10. Abel equation of the first kind
4.11. Abel equation of the second kind
4.12. Equation in total differentials
4.13. Integrating factor
4.14. (formula), "integration by differentiation"
4.15. (formula)
4.16. (formula)
4.17. (formula)
4.18. Clairaut's equations
4.19. Lagrange - d'Alembert equation
4.20. (formula). Legendre transformation
Chapter II. Arbitrary systems of differential equations solved with respect to derivatives
§ 5. Basic concepts
5.1. Notation and geometric meaning of the system of differential equations
5.2. Existence and uniqueness of a solution
5.3. Carathéodory's existence theorem
5.4. Dependence of the solution on the initial conditions and on the parameters
5.5. Sustainability Issues
§ 6. Solution methods
6.1. Polyline method
6.2. Picard-Lindelöf method of successive approximations
6.3. Application of power series
6.4. Relationship with partial differential equations
6.5. System reduction using a known relation between solutions
6.6. System reduction by differentiation and elimination
6.7. Estimate theorems
§ 7. Autonomous systems
7.1. Definition and geometric meaning of an autonomous system
7.2. On the behavior of integral curves in a neighborhood of a singular point in the case n = 2
7.3. Criteria for determining the type of singular point
Chapter III. Systems of linear differential equations
§ 8. Arbitrary linear systems
8.1. General remarks
8.2. Existence and uniqueness theorems. Solution Methods
8.3. Reduction of an inhomogeneous system to a homogeneous one
8.4. Estimate theorems
§ 9. Homogeneous linear systems
9.1. Solution properties. Fundamental Solution Systems
9.2. Existence theorems and solution methods
9.3. System reduction to a system with fewer equations
9.4. Conjugate system of differential equations
9.5. Self-adjoint systems of differential equations
9.6. Conjugate systems of differential forms; Lagrange identity, Green's formula
9.7. Fundamental Solutions
§ 10. Homogeneous linear systems with singular points
10.1. Singular Point Classifications
10.2. Weak singular points
10.3. Strong singular points
§ 11. Behavior of solutions for large values ​​of x
§ 12. Linear systems depending on a parameter
§ 13. Linear systems with constant coefficients
13.1. Homogeneous systems
13.2. More general systems
Chapter IV. Arbitrary differential equations of the nth order
§ 14. Equations resolved with respect to the highest derivative: (formula)
§ 15. Equations not resolved with respect to the highest derivative: (formula)
15.1. Equations in Total Differentials
15.2. Generalized homogeneous equations
15.3. Equations not explicitly containing x or y
Chapter V. Linear differential equations of the nth order
§ 16. Arbitrary linear differential equations of the nth order
16.1. General remarks
16.2. Existence and uniqueness theorems. Solution Methods
16.3. (n-1)-th Order Derivative Elimination
16.4. Reduction of an inhomogeneous differential equation to a homogeneous one
16.5. Behavior of Solutions for Large Values ​​of x
§ 17. Homogeneous linear differential equations of the nth order
17.1. Properties of Solutions and Existence Theorems
17.2. Reducing the order of a differential equation
17.3. On zeros of solutions
17.4. Fundamental Solutions
17.5. Conjugate, self-adjoint, and anti-self-adjoint differential forms
17.6. Lagrange identity; Dirichlet and Green's formulas
17.7. On solutions of adjoint equations and equations in total differentials
§ 18. Homogeneous linear differential equations with singular points
18.1. Classification of singular points
18.2. The case when the point (?) is regular or weakly singular
18.3. The case when the point (?) is regular or weakly singular
18.4. The case when the point (?) is strongly singular
18.5. The case when the point (?) is strongly singular
18.6. Differential Equations with Polynomial Coefficients
18.7. Differential Equations with Periodic Coefficients
18.8. Differential Equations with Doubly Periodic Coefficients
18.9. Real Variable Case
§ 19. Solution of linear differential equations using definite integrals
19.1. General principle
19.2. Laplace transform
19.3. Special Laplace transform
19.4. Mellin transform
19.5. Euler transform
19.6. Solution using double integrals
§ 20. Behavior of solutions for large values ​​of x
20.1. Polynomial Coefficients
20.2. More general coefficients
20.3. Continuous odds
20.4. Oscillation theorems
§ 21. Linear differential equations of the nth order, depending on the parameter
§ 22. Some special types of linear differential equations of the nth order
22.1. Homogeneous differential equations with constant coefficients
22.2. Inhomogeneous differential equations with constant coefficients
22.3. Euler equations
22.4. Laplace equation
22.5. Equations with polynomial coefficients
22.6. Pochhammer equation
Chapter VI. Second order differential equations
§ 23. Nonlinear differential equations of the second order
23.1. Methods for solving particular types of nonlinear equations
23.2. Some additional notes
23.3. Limit value theorems
23.4. Oscillation theorem
§ 24. Arbitrary second-order linear differential equations
24.1. General remarks
24.2. Some solution methods
24.3. Estimate theorems
§ 25. Homogeneous second-order linear differential equations
25.1. Reduction of second-order linear differential equations
25.2. Further Remarks on the Reduction of Second-Order Linear Equations
25.3. Expanding the Solution into a Continued Fraction
25.4. General remarks about solution zeros
25.5. Zeros of solutions on a finite interval
25.6. Behavior of solutions for (?)
25.7. Second Order Linear Differential Equations with Singular Points
25.8. Approximate solutions. Asymptotic solutions; real variable
25.9. Asymptotic solutions; complex variable
25.10. WBC method
Chapter VII. Linear differential equations of the third and fourth orders
§ 26. Linear differential equations of the third order
§ 27. Linear differential equations of the fourth order
Chapter VIII. Approximate methods for integrating differential equations
§ 28. Approximate integration of differential equations of the first order
28.1. Polyline method
28.2. Additional Half Step Method
28.3. Runge-Hein-Kutta method
28.4. Combining interpolation and successive approximations
28.5. Adams method
28.6. Additions to the Adams Method
§ 29. Approximate integration of higher-order differential equations
29.1. Approximate Integration Methods for Systems of First-Order Differential Equations
29.2. The broken line method for second-order differential equations
29.3. Runge*-Kutta method for differential equations of this order
29.4. Adams - Stoermer method for equation (formula)
29.5. Adams - Stoermer method for equation (formula)
29.6. Bless's method for equation (formula)
PART TWO
Boundary value and eigenvalue problems
Chapter I. Boundary value and eigenvalue problems for linear differential equations of the nth order
§ 1. General theory of boundary value problems
1.1. Notation and Preliminaries
1.2. Conditions for the solvability of a boundary value problem
1.3. Conjugate boundary value problem
1.4. Self-adjoint boundary value problems
1.5. Green's function
1.6. Solving an inhomogeneous boundary value problem using the Green's function
1.7. Generalized Green's function
§ 2. Boundary value problems and eigenvalue problems for an equation (formula)
2.1. Eigenvalues ​​and eigenfunctions; characteristic determinant (?)
2.2. Adjoint problem on eigenvalues ​​and the Greya resolvent; complete biorthogonal system
2.3. Normalized boundary conditions; regular eigenvalue problems
2.4. Eigenvalues ​​for regular and irregular eigenvalue problems
2.5. Expansion of a Given Function in Eigenfunctions of Regular and Irregular Eigenvalue Problems
2.6. Self-adjoint normal eigenvalue problems
2.7. On Integral Equations of Fredholm Type
2.8. Connection between boundary value problems and integral equations of Fredholm type
2.9. Relationship between eigenvalue problems and integral equations of Fredholm type
2.10. On integral equations of Volterra type
2.11. Relationship between boundary value problems and integral equations of Volterra type
2.12. Relationship between eigenvalue problems and integral equations of Volterra type
2.13. Relationship between eigenvalue problems and the calculus of variations
2.14. Application to expansion in terms of eigenfunctions
2.15. Additional Notes
§ 3. Approximate methods for solving eigenvalue problems and boundary value problems
3.1. Approximate Galerkin-Ritz method
3.2. Approximate Grammel Method
3.3. Solving an inhomogeneous boundary value problem using the Galerkin-Ritz method
3.4. Method of successive approximations
3.5. Approximate solution of boundary value problems and eigenvalue problems by the method of finite differences
3.6. Perturbation method
3.7. Eigenvalue estimates
3.8. Overview of ways to calculate eigenvalues ​​and eigenfunctions
§ 4. Self-adjoint eigenvalue problems for an equation (formula)
4.1. Formulation of the problem
4.2. General Preliminaries
4.3. Normal eigenvalue problems
4.4. Positive definite eigenvalue problems
4.5. Decomposition in terms of eigenfunctions
§ 5. Boundary and additional conditions of a more general form
Chapter II. Boundary Value Problems and Eigenvalue Problems for Systems of Linear Differential Equations
§ 6. Boundary value problems and eigenvalue problems for systems of linear differential equations
6.1. Notation and solvability conditions
6.2. Conjugate boundary value problem
6.3. Green's matrix
6.4. Eigenvalue problems
6.5. Self-adjoint eigenvalue problems
Chapter III. Boundary Value Problems and Eigenvalue Problems for Low-Order Equations
§ 7. Problems of the first order
7.1. Linear problems
7.2. Nonlinear Problems
§ 8. Linear boundary value problems of the second order
8.1. General remarks
8.2. Green's function
8.3. Estimates for solutions of boundary value problems of the first kind
8.4. Boundary conditions at (?)
8.5. Finding Periodic Solutions
8.6. One boundary value problem related to the study of fluid flow
§ 9. Linear eigenvalue problems of the second order
9.1. General remarks
9.2 Self-adjoint eigenvalue problems
9.3. (formula) and boundary conditions are self-adjoint
9.4. Eigenvalue problems and the variational principle
9.5. On the practical calculation of eigenvalues ​​and eigenfunctions
9.6. Eigenvalue problems, not necessarily self-adjoint
9.7. Additional conditions of a more general form
9.8. Eigenvalue Problems Containing Multiple Parameters
9.9. Differential Equations with Singularities at Boundary Points
9.10. Eigenvalue problems on an infinite interval
§ 10. Nonlinear boundary value problems and second-order eigenvalue problems
10.1. Boundary Value Problems for a Finite Interval
10.2. Boundary Value Problems for a Semi-Bounded Interval
10.3. Eigenvalue problems
§ 11. Boundary value problems and eigenvalue problems of the third - eighth orders
11.1. Linear eigenvalue problems of the third order
11.2. Linear eigenvalue problems of the fourth order
11.3. Linear problems for a system of two second-order differential equations
11.4. Nonlinear boundary value problems of the fourth order
11.5. Higher order eigenvalue problems
PART THREE SEPARATE DIFFERENTIAL EQUATIONS
Preliminary remarks
Chapter I. First Order Differential Equations
1-367. First degree differential equations with respect to (?)
368-517. Second degree differential equations with respect to (?)
518-544. Third degree differential equations with respect to (?)
545-576. Differential equations of a more general form
Chapter II. Second Order Linear Differential Equations
1-90. (formula)
91-145. (formula)
146-221. (formula)
222-250. (formula)
251-303. (formula)
304-341. (formula)
342-396. (formula)
397-410. (formula)
411-445. Other differential equations
Chapter III. Linear differential equations of the third order
Chapter IV. Linear differential equations of the fourth order
Chapter V. Linear Differential Equations of the Fifth and Higher Orders
Chapter VI. Nonlinear Second Order Differential Equations
1-72. (formula)
73-103. (formula)
104-187. (formula)
188-225. (formula)
226-249. Other differential equations
Chapter VII. Nonlinear differential equations of the third and higher orders
Chapter VIII. Systems of linear differential equations
Preliminary remarks
1-18. Systems of two differential equations of the first order with constant coefficients
19-25. Systems of two differential equations of the first order with variable coefficients
26-43. Systems of two differential equations of order higher than the first
44-57. Systems of more than two differential equations
Chapter IX. Systems of nonlinear differential equations
1-17. Systems of two differential equations
18-29. Systems of more than two differential equations
ADDITIONS
On the solution of linear homogeneous equations of the second order (I. Zbornik)
Additions to the book by E. Kamke (D. Mitrinovich)
A new way to classify linear differential equations and construct their general solution using recursive formulas (I. Zbornik)
Subject index

Ains E.L. Ordinary differential equations. Kharkov: ONTI, 1939

Andronov A.A., Leontovich E.V., Gordon I.I., Mayer A.G. Qualitative theory of dynamical systems of the second order. Moscow: Nauka, 1966

Anosov D.V. (ed.) Smooth dynamical systems (Collection of translations, Mathematics in foreign science N4). M.: Mir, 1977

Arnold V.I., Kozlov V.V., Neishtadt A.I. Mathematical aspects of classical and celestial mechanics. M.: VINITI, 1985

Barbashin E.A. Lyapunov functions. Moscow: Nauka, 1970

Bogolyubov N.N., Mitropolsky Yu.A. Asymptotic Methods in the Theory of Nonlinear Oscillations (2nd ed.). Moscow: Nauka, 1974

Vazov V. Asymptotic expansions of solutions of ordinary differential equations. M.: Mir, 1968

Weinberg M.M., Trenogin V.A. Theory of branching of solutions of nonlinear equations. Moscow: Nauka, 1969

Golubev V.V. Lectures on the analytical theory of differential equations. M.-L.: Gostekhteorizdat, 1950

Gursa E. Course of mathematical analysis, volume 2, part 2. Differential equations. M.-L.: GTTI, 1933

Demidovich B.P. Lectures on the mathematical theory of stability. Moscow: Nauka, 1967

Dobrovolsky V.A. Essays on the development of the analytical theory of differential equations. Kyiv: Vishcha school, 1974

Egorov D. Integration of differential equations (3rd ed.). M.: Printing Yakovlev, 1913

Erugin N.P. A Reading Book for a General Course in Differential Equations (3rd ed.). Minsk: Science and technology, 1979

Erugin N.P. Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients. Minsk: AN BSSR, 1963

Erugin N.P. Lappo-Danilevsky method in the theory of linear differential equations. L.: Leningrad State University, 1956

Zaitsev V.F. Introduction to modern group analysis. Part 1: Groups of transformations on the plane (textbook for the special course). St. Petersburg: Russian State Pedagogical University im. A.I. Herzen, 1996

Zaitsev V.F. Introduction to modern group analysis. Part 2: Equations of the first order and the point groups allowed by them (textbook for the special course). St. Petersburg: Russian State Pedagogical University im. A.I. Herzen, 1996

Ibragimov N.Kh. ABC of group analysis. Moscow: Knowledge, 1989

Ibragimov N.Kh. Experience of group analysis of ordinary differential equations. Moscow: Knowledge, 1991

Kamenkov G.V. Selected works. T.1. Movement stability. Fluctuations. Aerodynamics. Moscow: Nauka, 1971

Kamenkov G.V. Selected works. T.2. Stability and oscillations of nonlinear systems. Moscow: Nauka, 1972

Kamke E. Handbook of Ordinary Differential Equations (4th edition). Moscow: Nauka, 1971

Kaplansky I. Introduction to differential algebra. M.: IL, 1959

Kartashev A.P., Rozhdestvensky B.L. Ordinary Differential Equations and Fundamentals of the Calculus of Variations (2nd ed.). Moscow: Nauka, 1979

Coddington EA, Levinson N. Theory of ordinary differential equations. M.: IL, 1958

Kozlov V.V. Symmetries, topology and resonances in Hamiltonian mechanics. Izhevsk: Publishing House of the Udmurt State. university, 1995

Collatz L. Eigenvalue problems (with technical applications). Moscow: Nauka, 1968

Cole J. Perturbation methods in applied mathematics. M.: Mir, 1972

Koyalovich B.M. Research about the differential equation ydy-ydx=Rdx. St. Petersburg: Academy of Sciences, 1894

Krasovsky N.N. Some problems of the theory of motion stability. Moscow: Fizmatlit, 1959

Kruskal M. Adiabatic invariants. Asymptotic theory of Hamilton equations and other systems of differential equations, all solutions of which are approximately periodic. M.: IL, 1962

Kurensky M.K. Differential equations. Book 1. Ordinary differential equations. L .: Artillery Academy, 1933

Lappo-Danilevsky I.A. Application of functions from matrices to the theory of linear systems of ordinary differential equations. M.: GITTL, 1957

Lappo-Danilevsky I.A. Theory of functions from matrices and systems of linear differential equations. L.-M., GITTL, 1934

LaSalle J., Lefschetz S. Study of stability by the direct Lyapunov method. M.: Mir, 1964

Levitan B.M., Zhikov V.V. Almost periodic functions and differential equations. Moscow: Moscow State University, 1978

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