Ordinary Differential Equations Pdf For Engineering Mathematics

The Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, is an exceptional and complete reference for scientists and engineers as it contains over 7,000 ordinary differential equations with solutions. This book contains more equations and methods used in the field than any other book currently available. Included in the handbook are exact, asymptotic, approximate analytical, numerical symbolic and qualitative methods that are used for solving and analyzing linear and nonlinear equations. The authors also present formulas for effective construction of solutions and many different equations arising in various applications like heat transfer, elasticity, hydrodynamics and more. This extensive handbook is the perfect resource for engineers and scientists searching for an exhaustive reservoir of information on ordinary differential equations.

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HANDBOOK OF

ORDINARY

DIFFERENTIAL

EQUATIONS:

EXACT SOLUTIONS,

METHODS, AND

PROBLEMS

Andrei D. Polyanin

Valentin F. Zaitsev

PREFACE

The Handbook of Ordinary Differential Equations for Scientists and Engineers, is a

unique reference for scientists and engineers, which contains over 7,000 ordinary differ-

ential equations with solutions, as well as exact, asymptotic, approximate analytical, nu-

merical, symbolic, and qualitative methods for solving and analyzing linear and nonlinear

equations. First-, second-, third-, fourth- and higher-order ordinary differential equations

and systems of equations are considered. A number of new nonlinear equations, exact solu-

tions, transformations, and methods are described. Equations arising in various applications

(in the theory of heat and mass transfer, nonlinear mechanics, elasticity, hydrodynamics,

theory of nonlinear oscillations, combustion theory, chemical engineering science, etc.)

are considered. Analytical formulas for the effective construction of solutions are given.

Special attention is paid to equations of general form that depend on arbitrary functions.

Almost all other equations contain one or more arbitrary parameters (i.e., in fact, this book

deals with whole families of ordinary differential equations), which can be ﬁxed by the

reader at will. A number of speciﬁc examples where the methods described in the book are

used are considered. Statements of existence and uniqueness theorems as well as theorems

of stability and instability of solutions are given as well. Boundary-value problems and

eigenvalue problems are described. Signiﬁcant attention is given to Cauchy problems with

blow-up solutions as well as the important questions of nonexistence and nonuniqueness of

solutions to nonlinear boundary-value problems. Elements of bifurcation theory, Lie group

and discrete-group methods for ODEs, and the factorization principle are discussed. Sym-

bolic and numerical methods for solving ODEs problems with Maple, Mathematica, and

MATLAB are considered.

All in all, the handbook contains much more ordinary differential equations, problems,

methods, solutions, and transformations than any other book currently available. It essen-

tial that symbolic computation systems, even the most powerful ones such as Maple or

Mathematica, can provide no more than 40–50% of the exact analytical solutions to ODEs

given in this book (Chapters 13 through 18).

The main material is followed by a number of supplements, which present tables of

integrals, ﬁnite and inﬁnite series, and integral transforms as well as a brief description of

the basic properties of elementary and special functions (Bessel, modiﬁed Bessel, hyperge-

ometric, Legendre, etc.).

New material compared to Handbook of Exact Solutions for Ordinary Differential

Equations, 2003:

•The total volume of the new handbook has almost doubled (increased by nearly 700

pages).

•Some ﬁrst-, second-, and third-order nonlinear ODEs with solutions.

•Some analytical methods (including new methods) and standard numerical methods.

•Special numerical methods (including new methods) for solving problems with qual-

itative features or singularities.

•Symbolic and numerical methods with Maple, Mathematica, and MATLAB.

xxxiii

xxxiv P REFAC E

•Many new problems, illustrative examples, and ﬁgures.

•Elementary theory of using invariants for solving equations.

•Methods for the construction of particular solutions (including the method of differ-

ential constraints).

•Systems of coupled ordinary differential equations with solutions.

•Equations deﬁned parametrically or implicitly (exact and numerical methods and

exact solutions) as well as overdetermined systems of ODEs and underdetermined

ODEs.

For the convenience of a wide audience with varying mathematical backgrounds, the

authors tried to do their best to avoid special terminology whenever possible. Therefore,

some of the methods are outlined in a schematic and somewhat simpliﬁed manner, with

necessary references made to books where these methods are considered in more detail.

Many sections were written so that they could be read independently (moreover, many top-

ics do not require special mathematical background for their understanding and successful

practical application). This allows the reader to get to the heart of the matter quickly.

The handbook consists of parts, chapters, sections, subsections, and paragraphs. The

material within sections is arranged in increasing order of complexity. An extensive table

of contents and detailed index provides rapid access to the desired equations.

Isolated sections of the book can be used by university and college lecturers in practical

courses and lectures on ordinary differential equations for graduate and postgraduate stu-

dents. Furthermore, the second part of the book (Chapters 13–18) can be used as a database

of test problems for numerical, approximate analytical, and symbolic methods for solving

ordinary differential equations.

We would like to express our keen gratitude to Alexei Zhurov for fruitful discussions

and valuable remarks. We are very thankful to Inna Shingareva and Carlos Liz´

arraga-

Celaya, who wrote three chapters (19–21) of the book at our request. Also, we would like

to express our deep gratitude to Vladimir Nazaikinskii for translating several chapters of

this handbook.

The authors hope that the handbook will prove helpful for a wide audience of re-

searchers, university and college teachers, engineers, and students in various ﬁelds of math-

ematics, physics, mechanics, control, chemistry, economics, and engineering sciences.

Andrei D. Polyanin

Valentin F. Zaitsev

CONTENTS

Preface xxiii

Authors xxv

Basic Notation and Remarks xxvii

Part I. Methods for Ordinary Differential Equations 1

1 Methods for First-Order Differential Equations 3

1.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . . 3

1.1.1 Equations Solved for the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Equations Not Solved for the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Equations Solved for the Derivative. Simplest Techniques of Integration . . . . . . 8

1.2.1 Equations with Separable Variables and Related Equations . . . . . . . . . . . . 8

1.2.2 Homogeneous and Generalized Homogeneous Equations . . . . . . . . . . . . . 9

1.2.3 Linear Equation and Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 Darboux Equation and Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Exact Differential Equations. Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 General Riccati Equation. Simplest Integrable Cases. Polynomial

Solutions ....................................................... 13

1.4.2 Use of Particular Solutions to Construct the General Solution . . . . . . . . . 15

1.4.3 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.4 Special Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Abel Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.1 General Form of Abel Equations of the First Kind. Simplest Integrable

Cases .......................................................... 18

1.5.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6.1 General Form of Abel Equations of the Second Kind. Simplest Integrable

Cases .......................................................... 20

1.6.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6.3 Use of Particular Solutions to Construct Self-Transformations and the

General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.7 Classiﬁcation and Speciﬁc Features of Some Classes of Solutions . . . . . . . . . . . . 25

1.7.1 Stable and Unstable Solutions. Equilibrium Points . . . . . . . . . . . . . . . . . . . 25

1.7.2 Blow-Up Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.7.3 Space Localization of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.7.4 Cauchy Problems Admitting Non-Unique Solutions . . . . . . . . . . . . . . . . . . 36

1.8 Equations Not Solved for the Derivative and Equations Deﬁned Parametrically 37

1.8.1 Method of "Integration by Differentiation" for Equations Not Solved for

the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.8.2 Equations Not Solved for the Derivative. Speciﬁc Equations . . . . . . . . . . 38

1.8.3 Equations Deﬁned Parametrically and Differential-Algebraic Equations 40

1.9 Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.9.1 General Form of Contact Transformations. Method for the Construction

of Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.9.2 Examples of Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.10 Pfafﬁan Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.10.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.10.2 Completely Integrable Pfafﬁan Equations . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.10.3 Pfafﬁan Equations Not Satisfying the Integrability Condition . . . . . . . . 48

1.11 Approximate Analytic Methods for Solution of ODEs . . . . . . . . . . . . . . . . . . . . . 49

1.11.1 Method of Successive Approximations (Picard Method) . . . . . . . . . . . . 49

1.11.2 Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.11.3 Method of Series Expansion in the Independent Variable . . . . . . . . . . . 53

1.11.4 Method of Regular Expansion in the Small Parameter . . . . . . . . . . . . . . 56

1.12 Differential Inequalities and Solution Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1.12.1 Two Theorems on Solution Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1.12.2 Chaplygin's Theorem and Its Applications (Bilateral Estimates of the

Cauchy Problem Solution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.13 Standard Numerical Methods for Solving Ordinary Differential Equations . . . 61

1.13.1 Single-Step Methods. Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . 61

1.13.2 Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1.13.3 Predictor–Corrector Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

1.13.4 Modiﬁed Multistep Methods (Butcher's Methods) . . . . . . . . . . . . . . . . . 72

1.13.5 Stability and Convergence of Numerical Methods . . . . . . . . . . . . . . . . . . 72

1.13.6 Well- and Ill-Conditioned Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1.14 Special Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

1.14.1 Special Methods Based on Auxiliary Equations . . . . . . . . . . . . . . . . . . . 74

1.14.2 Numerical Integration of Equations That Contain Fixed Singular

Points ....................................................... 76

1.14.3 Numerical Integration of Equations Deﬁned Parametrically or

Implicitly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

1.14.4 Numerical Solution of Blow-Up Problems . . . . . . . . . . . . . . . . . . . . . . . . 81

1.14.5 Numerical Solution of Problems with Root Singularity . . . . . . . . . . . . . 89

2 Methods for Second-Order Linear Differential Equations 93

2.1 Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.1.1 Formulas for the General Solution. Wronskian Determinant . . . . . . . . . . . 93

2.1.2 Factorization and Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.2 Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.2.1 Existence Theorem. Kummer–Liouville Transformation . . . . . . . . . . . . . . 96

2.2.2 Formulas for the General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.3 Representation of Solutions as a Series in the Independent Variable . . . . . . . . . . 97

2.3.1 Equation Coefﬁcients are Representable in the Ordinary Power Series

Form .......................................................... 97

2.3.2 Equation Coefﬁcients Have Poles at Some Point . . . . . . . . . . . . . . . . . . . . . 98

2.4 Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.4.1 Equations Not Containing y ′

x..................................... 100

2.4.2 Equations Containing y ′

x......................................... 102

2.5 Boundary Value Problems. Green's Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.5.1 First, Second, Third, and Some Other Boundary Value Problems . . . . . . . 103

2.5.2 Simpliﬁcation of Boundary Conditions. Self-Adjoint Form of Equations 106

2.5.3 Green's and Modiﬁed Green's Functions. Representation Solutions via

Green's or Modiﬁed Green's Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.6 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.6.1 Sturm–Liouville Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.6.2 General Properties of the Sturm–Liouville Problem (2.6.1.1), (2.6.1.2) . 111

2.6.3 Problems with Boundary Conditions of the First Kind . . . . . . . . . . . . . . . . 112

2.6.4 Problems with Boundary Conditions of the Second Kind . . . . . . . . . . . . . 113

2.6.5 Problems with Boundary Conditions of the Third Kind . . . . . . . . . . . . . . . 114

2.6.6 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.7 Theorems on Estimates and Zeros of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.7.1 Theorem on Estimates of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.7.2 Sturm Comparison Theorem on Zeros of Solutions . . . . . . . . . . . . . . . . . . 115

2.7.3 Qualitative Behavior of Solutions as x → ∞ ........................ 116

2.8 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.8.1 Numerov's Method (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.8.2 Modiﬁed Shooting Method (Boundary Value Problems) . . . . . . . . . . . . . . 117

2.8.3 Sweep Method (Boundary Value Problems) . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.8.4 Method of Accelerated Convergence in Eigenvalue Problems . . . . . . . . . 119

2.8.5 Well-Conditioned and Ill-Conditioned Problems . . . . . . . . . . . . . . . . . . . . . 120

3 Methods for Second-Order Nonlinear Differential Equations 123

3.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . . 123

3.1.1 Equations Solved for the Derivative. General Solution . . . . . . . . . . . . . . . . 123

3.1.2 Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . 123

3.2 Some Transformations. Equations Admitting Reduction of Order . . . . . . . . . . . . 124

3.2.1 Equations Not Containing y or x Explicitly. Related Equations . . . . . . . . 124

3.2.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.2.3 Generalized Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.2.4 Equations Invariant under Scaling–Translation Transformations . . . . . . . 126

3.2.5 Exact Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.2.6 Nonlinear Equations Involving Linear Homogeneous Differential Forms 128

3.2.7 Reduction of Quasilinear Equations to the Normal Form . . . . . . . . . . . . . . 129

3.2.8 Equations Deﬁned Parametrically and Differential-Algebraic Equations 129

3.3 Boundary Value Problems. Uniqueness and Existence Theorems. Nonexistence

Theorems ............................................................ 133

3.3.1 Uniqueness and Existence Theorems for Boundary Value Problems . . . . 134

3.3.2 Reduction of Boundary Value Problems to Integral Equations. Integral

Identity. Jentzch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.3.3 Theorem on Nonexistence of Solutions to the First Boundary Value

Problem. Theorems on Existence of Two Solutions . . . . . . . . . . . . . . . . . . 139

3.3.4 Examples of Existence, Nonuniqueness, and Nonexistence of Solutions

to First Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.3.5 Theorems on Nonexistence of Solutions for the Mixed Problem.

Theorems on Existence of Two Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.3.6 Examples of Existence, Nonuniqueness, and Nonexistence of Solutions

to Mixed Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.3.7 Theorems on Existence of Two Solutions for the Third Boundary Value

Problem ........................................................ 151

3.3.8 Boundary Value Problems for Linear Equations with Nonlinear

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.4 Method of Regular Series Expansions with Respect to the Independent Variable 154

3.4.1 Method of Expansion in Powers of the Independent Variable . . . . . . . . . . 154

3.4.2 Pad´

e Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.5 Movable Singularities of Solutions of Ordinary Differential Equations. Painlev´

Equations ............................................................ 157

3.5.1 Preliminary Remarks. Singular Points of Solutions . . . . . . . . . . . . . . . . . . 157

3.5.2 First Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.5.3 Second Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.5.4 Third Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

3.5.5 Fourth Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

3.5.6 Fifth Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.5.7 Sixth Painlev´

e Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

3.6 Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.6.1 Preliminary Remarks. Summary Table of Basic Methods . . . . . . . . . . . . . 171

3.6.2 Method of Regular (Direct) Expansion in Powers of the Small Parameter 173

3.6.3 Method of Scaled Parameters (Lindstedt–Poincar ´

e Method) . . . . . . . . . . . 174

3.6.4 Averaging Method (Van der Pol–Krylov–Bogolyubov Scheme) . . . . . . . . 175

3.6.5 Method of Two-Scale Expansions (Cole–Kevorkian Scheme) . . . . . . . . . 177

3.6.6 Method of Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . 178

3.7 Galerkin Method and Its Modiﬁcations (Projection Methods) . . . . . . . . . . . . . . . . 181

3.7.1 Approximate Solution for a Boundary Value Problem . . . . . . . . . . . . . . . . 181

3.7.2 Galerkin Method. General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

3.7.3 Bubnov–Galerkin, Moment, and Least Squares Methods . . . . . . . . . . . . . . 182

3.7.4 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

3.7.5 Method of Partitioning the Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.7.6 Least Squared Error Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

3.8 Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

3.8.1 Method of Successive Approximations (Cauchy Problem) . . . . . . . . . . . . 185

3.8.2 Runge–Kutta Method (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

3.8.3 Reduction to a System of Equations (Cauchy Problem) . . . . . . . . . . . . . . . 186

3.8.4 Predictor–Corrector Methods (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . . 186

3.8.5 Shooting Method (Boundary Value Problems) . . . . . . . . . . . . . . . . . . . . . . . 187

3.8.6 Numerical Methods for Problems with Equations Deﬁned Implicitly or

Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

3.8.7 Numerical Solution Blow-Up Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4 Methods for Linear ODEs of Arbitrary Order 197

4.1 Linear Equations with Constant Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

4.1.1 Homogeneous Linear Equations. General Solution . . . . . . . . . . . . . . . . . . . 197

4.1.2 Nonhomogeneous Linear Equations. General and Particular Solutions . . 198

4.2 Linear Equations with Variable Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.2.1 Homogeneous Linear Equations. General Solution. Order Reduction.

Liouville Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.2.2 Nonhomogeneous Linear Equations. General Solution. Superposition

Principle ....................................................... 201

4.2.3 Nonhomogeneous Linear Equations. Cauchy Problem. Reduction to

Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

4.3 Laplace Transform and the Laplace Integral. Applications to Linear ODEs . . . . 204

4.3.1 Laplace Transform and the Inverse Laplace Transform . . . . . . . . . . . . . . . 204

4.3.2 Main Properties of the Laplace Transform. Inversion Formulas for Some

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

4.3.3 Limit Theorems. Representation of Inverse Transforms as Convergent

Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.3.4 Solution of the Cauchy Problem for Constant-Coefﬁcient Linear ODEs.

Applications to Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . . 209

4.3.5 Solution of Linear Equations with Polynomial Coefﬁcients Using the

Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4.3.6 Solution of Linear Equations with Polynomial Coefﬁcients Using the

Laplace Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

4.4 Asymptotic Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.4.1 Fourth-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.4.2 Higher-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 214

4.5 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

4.5.1 Statement of the Problem. Approximate Solution . . . . . . . . . . . . . . . . . . . . 215

4.5.2 Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

5 Methods for Nonlinear ODEs of Arbitrary Order 217

5.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . . 217

5.1.1 Equations Solved for the Derivative. General Solution . . . . . . . . . . . . . . . . 217

5.1.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.2 Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.2.1 Equations Not Containing y or x Explicitly . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.2.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

5.2.3 Generalized Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.2.4 Equations Invariant under Scaling-Translation Transformations . . . . . . . . 220

5.2.5 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.3 Method for Construction of Solvable Equations of General Form . . . . . . . . . . . . 222

5.3.1 Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

5.3.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.4 Numerical Integration of n -order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5.4.1 Numerical Solution of the Cauchy Problem for n -order ODEs . . . . . . . . . 224

5.4.2 Numerical Solution of Equations Deﬁned Implicitly or Parametrically . . 224

6 Methods for Linear Systems of ODEs 227

6.1 Systems of Linear Constant-Coefﬁcient Equations . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.1.1 Systems of First-Order Linear Homogeneous Equations. General

Solution ........................................................ 227

6.1.2 Systems of First-Order Linear Homogeneous Equations. Particular

Solutions ....................................................... 228

6.1.3 Nonhomogeneous Systems of Linear First-Order Equations . . . . . . . . . . . 230

6.1.4 Homogeneous Linear Systems of Higher-Order Differential Equations . 231

6.1.5 Normal Coordinates and Natural Oscillations . . . . . . . . . . . . . . . . . . . . . . . 232

6.1.6 Nonhomogeneous Higher-Order Linear Systems. D'Alembert's Method 233

6.1.7 Usage of the Laplace Transform for Solving Linear Systems of

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

6.1.8 Classiﬁcation of Equilibrium Points of Two-Dimensional Linear

Systems ........................................................ 235

6.2 Systems of Linear Variable-Coefﬁcient Equations . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.2.1 Homogeneous Systems of Linear First-Order Equations . . . . . . . . . . . . . . 240

6.2.2 Nonhomogeneous Systems of Linear First-Order Equations . . . . . . . . . . . 242

6.2.3 Euler System of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . 243

7 Methods for Nonlinear Systems of ODEs 245

7.1 Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . . . 245

7.1.1 Systems Solved for the Derivative. A Solution and the General Solution 245

7.1.2 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.1.3 Reduction of Systems of Equations to a Single Equation or to an

Autonomous System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.1.4 First Integrals. Using Them to Reduce System Dimension . . . . . . . . . . . . 247

7.2 Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . . . 248

7.2.1 Integrable Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

7.2.2 Autonomous Systems and Their Reduction to Systems of Lower

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.3 Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

7.3.1 Lyapunov Stability. Asymptotic Stability. Unstable Solutions . . . . . . . . . 250

7.3.2 Theorems of Stability and Instability by First Approximation . . . . . . . . . 251

7.3.3 Lyapunov Function. Theorems of Stability and Instability . . . . . . . . . . . . 253

7.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.4.1 Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.4.2 Systems Involving Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . 259

8 Elements of Bifurcation Theory 263

8.1 Dynamical Systems. Rough and Nonrough Systems . . . . . . . . . . . . . . . . . . . . . . . 263

8.1.1 Bifurcation. Dynamical Systems. Phase Portrait . . . . . . . . . . . . . . . . . . . . . 263

8.1.2 Topologically Equivalent Systems. Rough and Nonrough Systems . . . . . 264

8.2 Bifurcations of Second-Order Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 265

8.2.1 Second-Order Dynamical Systems. Rough and Nonrough Systems . . . . . 265

8.2.2 Bifurcations in Systems of the First Degree of Nonroughness . . . . . . . . . 266

8.3 Bifurcations of Solutions to Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . 270

8.3.1 Bifurcations of Solutions to Linear Boundary Value Problems . . . . . . . . . 270

8.3.2 Bifurcations in Solutions to Nonlinear Boundary Value Problems . . . . . . 270

8.3.3 Bifurcation Analysis of Boundary Value Problems without Linearizing

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

9 Elementary Theory of Using Invariants for Solving Equations 277

9.1 Introduction. Symmetries. General Scheme of Using Invariants for Solving

Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

9.1.1 Symmetries. Transformations Preserving the Form of Equations.

Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

9.1.2 General Scheme of Using Invariants for Solving Mathematical

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9.2 Algebraic Equations and Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9.2.1 Algebraic Equations with Even Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9.2.2 Reciprocal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

9.2.3 Systems of Algebraic Equations Symmetric with Respect to Permutation

of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

9.3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

9.3.1 Transformations Preserving the Form of Equations. Invariants . . . . . . . . . 283

9.3.2 Order Reduction Procedure for Equations with n≥2 (Reduction to

Solvable Form with n = 1 ) ........................................ 284

9.3.3 Simple Transformations. Invariant Determination Procedure . . . . . . . . . . 284

9.3.4 Analysis of Some Ordinary Differential Equations. Useful Remarks . . . . 285

10 Methods for the Construction of Particular Solutions 289

10.1 Two Problems on Searching for Particular Solutions to ODEs with Parameters 289

10.1.1 Preliminary Remarks. Traveling Wave Solutions . . . . . . . . . . . . . . . . . . 289

10.1.2 Two Problems for ODEs with Parameters. Conditional Capacity of

Exact Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.2 Method of Undetermined Coefﬁcients and Its Special Cases . . . . . . . . . . . . . . . . 292

10.2.1 General Description of the Method of Undetermined Coefﬁcients . . . . 292

10.2.2 Power-Law, Tanh-Coth, and Sine-Cosine Methods . . . . . . . . . . . . . . . . . 293

10.2.3 Exp-Function, Q-Expansion and Related Methods . . . . . . . . . . . . . . . . . 297

10.3 Method of Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

10.3.1 Preliminary Remarks. First-Order Differential Constraints and Their

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

10.3.2 Differential Constraints of Arbitrary Order. General Consistency

Method for Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

10.3.3 Using Point Transformations in Combination with the Method of

Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.3.4 Using Several Differential Constraints. G′

/G-Expansion Method and

Simplest Equation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

11 Group Methods for ODEs 313

11.1 Lie Group Method. Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

11.1.1 Local One-Parameter Lie Group of Transformations. Invariance

Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

11.1.2 Group Analysis of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . 316

11.1.3 Utilization of Local Groups for Reducing the Order of Equations and

Their Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

11.1.4 Seeking Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

11.2 Contact and B¨

acklund Transformations. Formal Operators. Factorization

Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

11.2.1 Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

11.2.2 B¨

acklund Transformations. Formal Operators and Nonlocal Variables 323

11.2.3 Factorization Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

11.3 First Integrals (Conservation Laws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

11.3.1 Algorithm of Finding First Integrals of ODEs . . . . . . . . . . . . . . . . . . . . . 334

11.3.2 Applications to Second-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

11.3.3 Lie–B¨

acklund Symmetries Generated by First Integrals . . . . . . . . . . . . . 337

11.4 Underdetermined Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.2 Factorization Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

11.4.3 Some Technical Elements. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

11.4.4 On Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

12 Discrete-Group Methods 355

12.1 Discrete Group Method for Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . 355

12.1.1 Classes of ODEs with Parameters. Discrete Group of Point

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

12.1.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

12.2 Discrete Group Method Based on RF-Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

12.2.1 General Description of the Method. First and Second RF-Pairs . . . . . . 358

12.2.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

12.3 Discrete Group Method Based on the Inclusion Method . . . . . . . . . . . . . . . . . . . 364

Part II. Exact Solutions of Ordinary Differential Equations 365

13 First-Order Ordinary Differential Equations 367

13.1 Simplest Equations with Arbitrary Functions Integrable in Closed Form . . . . . 367

13.1.1 Equations of the Form y ′

x=f( x)................................ 367

13.1.2 Equations of the Form y ′

x=f( y)................................ 367

13.1.3 Separable Equations y ′

x=f(x ) g( y).............................. 367

13.1.4 Linear Equation g (x )y ′

x=f 1 (x) y+ f 0 (x)........................ 368

13.1.5 Bernoulli Equation g (x )y ′

x=f 1 (x) y+ f n (x)y n . . . . . . . . . . . . . . . . . . . 368

13.1.6 Homogeneous Equation y ′

x=f(y/x ) ............................ 368

13.2 Riccati Equation g (x)y ′

x=f 2 (x) y 2 +f 1 (x)y + f 0 (x)...................... 368

13.2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

13.2.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

13.2.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 375

13.2.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 378

13.2.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 380

13.2.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 382

13.2.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 387

13.2.8 Equations with Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

13.2.9 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

13.3 Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

13.3.1 Equations of the Form yy′

x−y= f( x)........................... 394

13.3.2 Equations of the Form yy′

x=f( x) y+1 .......................... 409

13.3.3 Equations of the Form yy′

x=f 1 (x)y + f 0 (x)...................... 410

13.3.4 Equations of the Form [g1 (x )y +g0 (x)]y ′

x=f 2 (x) y 2 +f 1 (x) y+ f 0 (x) 422

13.3.5 Some Types of First- and Second-Order Equations Reducible to Abel

Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

13.4 Equations Containing Polynomial Functions of y . . . . . . . . . . . . . . . . . . . . . . . . . 431

13.4.1 Abel Equations of the First Kind

y′

x=f 3 (x)y 3 +f 2 (x) y 2 +f 1 (x)y + f 0 (x). . . . . . . . . . . . . . . . . . . . . . . . . 431

13.4.2 Equations of the Form (A22 y2 +A 12 xy + A11 x2 + A0 )y ′

B22 y2 + B12 xy + B11x2 +B0 ................................... 436

13.4.3 Equations of the Form (A22 y2 +A 12 xy + A11 x2 + A2 y + A1 x )y ′

B22 y2 + B12 xy + B11x2 +B2 y+ B1 x........................... 438

13.4.4 Equations of the Form (A22 y2 + A12xy + A11 x2 + A2 y + A1 x + A0 )y ′

B22 y2 + B12 xy + B11x2 +B2 y+ B1 x+ B0 . . . . . . . . . . . . . . . . . . . . . . . 447

13.4.5 Equations of the Form (A3 y3 +A2 xy2 + A1x2 y + A0 x3 + a1 y +a0 x )y ′

B3 y3 + B2xy2 + B1 x2 y+ B0 x3 + b1 y+ b0 x ...................... 452

13.5 Equations of the Form f (x, y)y ′

x=g( x, y) Containing Arbitrary Parameters . 456

13.5.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

13.5.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 460

13.5.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 464

13.5.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 466

13.5.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 467

13.5.6 Equations Containing Combinations of Exponential, Hyperbolic,

Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 469

13.6 Equations of the Form F (x, y, y′

x) = 0 Containing Arbitrary Parameters . . . . . 471

13.6.1 Equations of the Second Degree in y ′

x............................ 471

13.6.2 Equations of the Third Degree in y ′

x............................. 478

13.6.3 Equations of the Form (y ′

x) k =f( y)+ g( x) . . . . . . . . . . . . . . . . . . . . . . . 481

13.6.4 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

13.7 Equations of the Form f (x, y)y ′

x=g( x, y) Containing Arbitrary Functions . . 495

13.7.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

13.7.2 Equations Containing Exponential and Hyperbolic Functions . . . . . . . 498

13.7.3 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 500

13.7.4 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 501

13.7.5 Equations Containing Combinations of Exponential, Logarithmic, and

Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

13.8 Equations Not Solved for the Derivative and Equations Deﬁned Parametrically 504

13.8.1 Equations Not Solved for the Derivative Containing Arbitrary

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

13.8.2 Some Transformations of Equations Not Solved for the Derivative . . . 514

13.8.3 Equations Deﬁned Parametrically Containing Arbitrary Functions . . . 515

14 Second-Order Ordinary Differential Equations 519

14.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

14.1.1 Representation of the General Solution through a Particular Solution . 519

14.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

14.1.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 553

14.1.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 560

14.1.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 565

14.1.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 568

14.1.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 580

14.1.8 Equations Containing Combinations of Exponential, Logarithmic,

Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

14.1.9 Equations with Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

14.1.10 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

14.2 Autonomous Equations y ′′

xx =F(y, y ′

x).................................. 605

14.2.1 Equations of the Form y ′′

xx −y ′

x=f( y)........................... 606

14.2.2 Equations of the Form y ′′

xx +f( y)y ′

x+y= 0 ...................... 610

14.2.3 Lienard Equations y ′′

xx +f( y)y ′

x+g( y) = 0 . . . . . . . . . . . . . . . . . . . . . . . 613

14.2.4 Rayleigh Equations y ′′

xx +f( y ′

x)+ g(y ) = 0 . . . . . . . . . . . . . . . . . . . . . . . 616

14.3 Emden–Fowler Equation y ′′

xx =Ax n y m ................................. 619

14.3.1 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

14.3.2 First Integrals (Conservation Laws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

14.3.3 Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

14.4 Equations of the Form y ′′

xx =A 1 x n 1 y m 1 +A 2 x n 2 y m 2 . . . . . . . . . . . . . . . . . . . . . 628

14.4.1 Classiﬁcation Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

14.4.2 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

14.5 Generalized Emden–Fowler Equation y ′′

xx =Ax n y m (y ′

x) l . . . . . . . . . . . . . . . . . . 652

14.5.1 Classiﬁcation Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

14.5.2 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

14.5.3 Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

14.6 Equations of the Form y ′′

xx =A 1 x n 1 y m 1 (y ′

x) l 1 +A 2 x n 2 y m 2 (y ′

x) l 2 . . . . . . . . . . . 678

14.6.1 Modiﬁed Emden–Fowler Equation y ′′

xx =A 1 x −1y ′

x+A 2 x n y m . . . . . . 678

14.6.2 Equations of the Form y ′′

xx = (A 1 x n 1 y m 1 +A 2 x n 2 y m 2 )(y ′

x) l . . . . . . . . 688

14.6.3 Equations of the Form y ′′

xx =σAx n y m (y ′

x) l +Ax n−1y m+1(y ′

x) l−1. . . 718

14.6.4 Other Equations (l1 ̸ =l2 ) ....................................... 733

14.7 Equations of the Form y ′′

xx =f( x) g( y)h ( y ′

x)............................. 739

14.7.1 Equations of the Form y ′′

xx =f(x ) g( y)........................... 739

14.7.2 Equations Containing Power Functions (h ̸≡ const) . . . . . . . . . . . . . . . . 742

14.7.3 Equations Containing Exponential Functions (h ̸≡ const ) . . . . . . . . . . . 746

14.7.4 Equations Containing Hyperbolic Functions (h ̸≡ const ) . . . . . . . . . . . . 750

14.7.5 Equations Containing Trigonometric Functions (h ̸≡ const ) . . . . . . . . . 751

14.7.6 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

14.8 Some Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 753

14.8.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

14.8.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 761

14.8.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 769

14.8.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 775

14.8.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 777

14.8.6 Equations Containing the Combinations of Exponential, Hyperbolic,

Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 783

14.9 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

14.9.1 Equations of the Form F (x, y)y ′′

xx +G(x, y) = 0 . . . . . . . . . . . . . . . . . . 786

14.9.2 Equations of the Form F (x, y)y ′′

xx +G(x, y) y ′

x+H( x, y) = 0 . . . . . . . 793

14.9.3 Equations of the Form F (x, y)y ′′

xx +∑ M

m=0 G m (x, y)(y ′

x) m = 0

(M = 2, 3, 4) ................................................. 798

14.9.4 Equations of the Form F (x, y, y′

x)y ′′

xx +G(x, y, y ′

x)=0 . . . . . . . . . . . . 802

14.9.5 Equations Not Solved for Second Derivative . . . . . . . . . . . . . . . . . . . . . . 813

14.9.6 Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

14.9.7 Equations Deﬁned Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822

14.9.8 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825

15 Third-Order Ordinary Differential Equations 829

15.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

15.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

15.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

15.1.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 848

15.1.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 853

15.1.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 863

15.1.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 866

15.1.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 879

15.1.8 Equations Containing Combinations of Exponential, Logarithmic,

Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884

15.1.9 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 892

15.2 Equations of the Form y ′′′

xxx =Ax α y β (y ′

x) γ (y ′′

xx) δ ........................ 901

15.2.1 Classiﬁcation Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901

15.2.2 Equations of the Form y ′′′

xxx =Ay β .............................. 909

15.2.3 Equations of the Form y ′′′

xxx =Ax α y β ............................ 911

15.2.4 Equations with |γ |+ |δ | ̸= 0 ..................................... 912

15.2.5 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944

15.3 Equations of the Form y ′′′

xxx =f( y) g( y ′

x)h(y ′′

xx).......................... 945

15.3.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 945

15.3.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 948

15.3.3 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952

15.4 Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

15.4.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

15.4.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 963

15.4.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 967

15.4.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 971

15.4.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 974

15.5 Nonlinear Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . 978

15.5.1 Equations of the Form F (x, y)y ′′′

xxx +G( x, y) = 0 . . . . . . . . . . . . . . . . . 978

15.5.2 Equations of the Form F (x, y, y′

x)y ′′′

xxx +G( x, y, y ′

x) = 0 . . . . . . . . . . . 980

15.5.3 Equations of the Form F (x, y, y′

x)y ′′′

xxx+G(x, y, y ′

x)y ′′

xx+H(x, y, y ′

x)=

0............................................................ 987

15.5.4 Equations of the Form F (x, y, y′

x)y ′′′

xxx + ∑ αG α (x, y, y ′

x)(y ′′

xx) α = 0 992

15.5.5 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994

16 Fourth-Order Ordinary Differential Equations 999

16.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999

16.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999

16.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000

16.1.3 Equations Containing Exponential and Hyperbolic Functions . . . . . . . 1007

16.1.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 1011

16.1.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1012

16.1.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 1015

16.2 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019

16.2.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019

16.2.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 1027

16.2.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 1029

16.2.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 1034

16.2.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1035

16.2.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 1040

17 Higher-Order Ordinary Differential Equations 1051

17.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

17.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

17.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

17.1.3 Equations Containing Exponential and Hyperbolic Functions . . . . . . . 1058

17.1.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 1061

17.1.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1061

17.1.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 1064

17.2 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068

17.2.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068

17.2.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 1075

17.2.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 1077

17.2.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 1081

17.2.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1083

17.2.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 1087

18 Some Systems of Ordinary Differential Equations 1099

18.1 Linear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099

18.1.1 Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099

18.1.2 Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

18.2 Linear Systems of Three and More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

18.3 Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

18.3.1 Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

18.3.2 Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110

18.4 Nonlinear Systems of Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1113

18.4.1 Systems of Three Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113

18.4.2 Dynamics of a Rigid Body with a Fixed Point . . . . . . . . . . . . . . . . . . . . . 1115

Part III. Symbolic and Numerical Solutions of Nonlinear PDEs with

Maple, Mathematica, and MATLAB 1119

19 Symbolic and Numerical Solutions of ODEs with Maple 1121

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121

19.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121

19.1.2 Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122

19.1.3 Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124

19.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127

19.2.1 Exact Analytical Solutions in Terms of Predeﬁned Functions . . . . . . . . 1127

19.2.2 Exact Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . 1137

19.2.3 Different Types of Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 1142

19.2.4 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 1144

19.2.5 Integral Transform Methods for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147

19.2.6 Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . . 1149

19.3 Group Analysis of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

19.3.1 Solution Strategies and Predeﬁned Functions . . . . . . . . . . . . . . . . . . . . . . 1153

19.3.2 Constructing Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155

19.3.3 Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156

19.3.4 Order Reduction of ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158

19.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160

19.4.1 Numerical Solutions in Terms of Predeﬁned Functions . . . . . . . . . . . . . 1160

19.4.2 Numerical Methods Embedded in Maple . . . . . . . . . . . . . . . . . . . . . . . . . 1162

19.4.3 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . . 1168

19.4.4 Initial Value Problems: Constructing Numerical Methods and

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171

19.4.5 Boundary Value Problems: Examples of Numerical Solutions . . . . . . . 1176

19.4.6 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . . 1181

19.4.7 First-Order Systems of ODEs. Higher-Order ODEs. Numerical

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183

20 Symbolic and Numerical Solutions of ODEs with Mathematica 1187

20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187

20.1.1 Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187

20.1.2 Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189

20.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

20.2.1 Exact Analytical Solutions in Terms of Predeﬁned Functions . . . . . . . . 1193

20.2.2 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . 1203

20.2.3 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 1207

20.2.4 Integral Transform Methods for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209

20.2.5 Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . . 1212

20.3 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215

20.3.1 Numerical Solutions in Terms of Predeﬁned Functions . . . . . . . . . . . . . 1215

20.3.2 Numerical Methods Embedded in Mathematica . . . . . . . . . . . . . . . . . . . 1217

20.3.3 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . . 1221

20.3.4 Initial Value Problems: Constructing Numerical Methods and

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223

20.3.5 Boundary Value Problems: Examples of Numerical Solutions . . . . . . . 1229

20.3.6 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . . 1235

20.3.7 First-Order Systems of ODEs. Higher-Order ODEs. Numerical

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236

20.3.8 Phase Plane Analysis for First-Order Autonomous Systems . . . . . . . . . 1239

20.3.9 Numerical-Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1241

21 Symbolic and Numerical Solutions of ODEs with MATLAB 1245

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245

21.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245

21.1.2 Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246

21.1.3 MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249

21.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253

21.2.1 Analytical Solutions in Terms of Predeﬁned Functions . . . . . . . . . . . . . 1253

21.2.2 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . 1259

21.2.3 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 1261

21.3 Numerical Solutions of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263

21.3.1 Numerical Solutions via Predeﬁned Functions . . . . . . . . . . . . . . . . . . . . . 1263

21.3.2 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . . 1268

21.3.3 Boundary Value Problems: Examples of Numerical Solutions . . . . . . . 1272

21.3.4 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . . 1276

21.4 Numerical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279

21.4.1 First-Order Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279

21.4.2 First-Order Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282

Part IV. Supplements 1285

S1 Elementary Functions and Their Properties 1287

S1.1 Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1287

S1.1.1 Properties of the Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287

S1.1.2 Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287

S1.1.3 Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288

S1.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289

S1.2.1 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289

S1.2.2 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290

S1.2.3 Relations between Trigonometric Functions of Single Argument . . . . . 1290

S1.2.4 Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . 1290

S1.2.5 Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

S1.2.6 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

S1.2.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

S1.2.8 Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . 1292

S1.2.9 Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . 1292

S1.2.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292

S1.2.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292

S1.2.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293

S1.2.13 Representation in the Form of Inﬁnite Products . . . . . . . . . . . . . . . . . . . 1293

S1.2.14 Euler and de Moivre Formulas. Relationship with Hyperbolic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293

S1.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293

S1.3.1 Deﬁnitions of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1293

S1.3.2 Simplest Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294

S1.3.3 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294

S1.3.4 Relations between Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 1295

S1.3.5 Addition and Subtraction of Inverse Trigonometric Functions . . . . . . . 1295

S1.3.6 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296

S1.3.7 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296

S1.3.8 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296

S1.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296

S1.4.1 Deﬁnitions of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296

S1.4.2 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297

S1.4.3 Relations between Hyperbolic Functions of Single Argument (x≥ 0 ) . 1297

S1.4.4 Addition and Subtraction of Hyperbolic Functions . . . . . . . . . . . . . . . . . 1297

S1.4.5 Products of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297

S1.4.6 Powers of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298

S1.4.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298

S1.4.8 Hyperbolic Functions of Multiple Argument . . . . . . . . . . . . . . . . . . . . . . 1298

S1.4.9 Hyperbolic Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

S1.4.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

S1.4.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

S1.4.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

S1.4.13 Representation in the Form of Inﬁnite Products . . . . . . . . . . . . . . . . . . . 1299

S1.4.14 Relationship with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1300

S1.5 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300

S1.5.1 Deﬁnitions of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 1300

S1.5.2 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300

S1.5.3 Relations between Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . 1300

S1.5.4 Addition and Subtraction of Inverse Hyperbolic Functions . . . . . . . . . . 1300

S1.5.5 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

S1.5.6 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

S1.5.7 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

S2 Indeﬁnite and Deﬁnite Integrals 1303

S2.1 Indeﬁnite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303

S2.1.1 Integrals Involving Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303

S2.1.2 Integrals Involving Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307

S2.1.3 Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 1310

S2.1.4 Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1311

S2.1.5 Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 1314

S2.1.6 Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1315

S2.1.7 Integrals Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . . 1319

S2.2 Tables of Deﬁnite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1320

S2.2.1 Integrals Involving Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1320

S2.2.2 Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 1323

S2.2.3 Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1324

S2.2.4 Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 1325

S2.2.5 Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1325

S2.2.6 Integrals Involving Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328

S3 Tables of Laplace and Inverse Laplace Transforms 1331

S3.1 Tables of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1331

S3.1.1 General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1331

S3.1.2 Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1333

S3.1.3 Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1333

S3.1.4 Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1334

S3.1.5 Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1335

S3.1.6 Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 1335

S3.1.7 Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337

S3.2 Tables of Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338

S3.2.1 General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338

S3.2.2 Expressions with Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1340

S3.2.3 Expressions with Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344

S3.2.4 Expressions with Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345

S3.2.5 Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1346

S3.2.6 Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1347

S3.2.7 Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1348

S3.2.8 Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 1349

S3.2.9 Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349

S4 Special Functions and Their Properties 1351

S4.1 Some Coefﬁcients, Symbols, and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351

S4.1.1 Binomial Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351

S4.1.2 Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352

S4.1.3 Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352

S4.1.4 Euler Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353

S4.2 Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . . 1354

S4.2.1 Error Function and Complementary Error Function . . . . . . . . . . . . . . . . 1354

S4.2.2 Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354

S4.2.3 Logarithmic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355

S4.3 Sine Integral and Cosine Integral. Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . 1356

S4.3.1 Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356

S4.3.2 Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357

S4.3.3 Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357

S4.4 Gamma Function, Psi Function, and Beta Function . . . . . . . . . . . . . . . . . . . . . . . 1358

S4.4.1 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358

S4.4.2 Psi Function (Digamma Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359

S4.4.3 Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1360

S4.5 Incomplete Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1360

S4.5.1 Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1360

S4.5.2 Incomplete Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1361

S4.6 Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362

S4.6.1 Deﬁnitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362

S4.6.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . 1364

S4.6.3 Zeros and Orthogonality Properties of Bessel Functions . . . . . . . . . . . . 1366

S4.6.4 Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . . 1367

S4.7 Modiﬁed Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

S4.7.1 Deﬁnitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

S4.7.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . 1369

S4.8 Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1370

S4.8.1 Deﬁnition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1370

S4.8.2 Power Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . 1370

S4.9 Degenerate Hypergeometric Functions (Kummer Functions) . . . . . . . . . . . . . . . 1371

S4.9.1 Deﬁnitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371

S4.9.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . 1374

S4.9.3 Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375

S4.10 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375

S4.10.1 Various Representations of the Hypergeometric Function . . . . . . . . 1375

S4.10.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377

S4.11 Legendre Polynomials, Legendre Functions, and Associated Legendre

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377

S4.11.1 Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . . 1377

S4.11.2 Associated Legendre Functions with Integer Indices and Real

Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1380

S4.11.3 Associated Legendre Functions. General Case . . . . . . . . . . . . . . . . . 1380

S4.12 Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383

S4.12.1 Deﬁnitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383

S4.12.2 Integral Representations, Asymptotic Expansions, and Linear

Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384

S4.13 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385

S4.13.1 Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385

S4.13.2 Incomplete Elliptic Integrals (Elliptic Integrals) . . . . . . . . . . . . . . . . 1386

S4.14 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388

S4.14.1 Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388

S4.14.2 Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392

S4.15 Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394

S4.15.1 Series Representation of the Jacobi Theta Functions. Simplest

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394

S4.15.2 Various Relations and Formulas. Connection with Jacobi Elliptic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395

S4.16 Mathieu Functions and Modiﬁed Mathieu Functions . . . . . . . . . . . . . . . . . . . . 1396

S4.16.1 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396

S4.16.2 Modiﬁed Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398

S4.17 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398

S4.17.1 Laguerre Polynomials and Generalized Laguerre Polynomials . . . . 1398

S4.17.2 Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . 1400

S4.17.3 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402

S4.17.4 Jacobi Polynomials and Gegenbauer Polynomials . . . . . . . . . . . . . . . 1404

S4.18 Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

S4.18.1 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

S4.18.2 Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406

References 1409

Index 1427

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... 2. If questions of the integrability of equation (2) are considered, then the values of the parameter are inessential, and the values of are essential. Currently, only two values = ± 6 25 are known, for which equation (2) admits a closed-form solution [27]. Therefore, if in some paper the integrability of this equation for other values of is proved, this result will certainly be new. ...

... Remark 4. Using handbooks [26,27], which contain many exact solutions of ordinary and partial differential equations, it is not difficult to give other examples of nonlinear equations that have qualitative features similar to Example 5. ...

The problems of estimating the similarity index of mathematical and other scientific publications containing equations and formulas are discussed for the first time. It is shown that the presence of equations and formulas (as well as figures, drawings, and tables) is a complicating factor that significantly complicates the study of such texts. It is shown that the method for determining the similarity index of publications, based on taking into account individual mathematical symbols and parts of equations and formulas, is ineffective and can lead to erroneous and even completely absurd conclusions. The possibilities of the most popular software system iThenticate, currently used in scientific journals, are investigated for detecting plagiarism and self-plagiarism. The results of processing by the iThenticate system of specific examples and special test problems containing equations and formulas are presented. It has been established that this software system when analyzing inhomogeneous texts, is often unable to distinguish self-plagiarism from pseudo-self-plagiarism (false self-plagiarism). A model complex situation is considered, in which the identification of self-plagiarism requires the involvement of highly qualified specialists of a narrow profile. Various ways to improve the work of software systems for comparing inhomogeneous texts are proposed. This article will be useful to researchers and university teachers in mathematics, physics, and engineering sciences, programmers dealing with problems in image recognition and research topics of digital image processing, as well as a wide range of readers who are interested in issues of plagiarism and self-plagiarism.

... Interested readers in the exact solutions of (2.12) for varieties of the functions a(x), p(x) and q(x), are referred to [24,25]. ...

A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized form of nonlinear equations of reaction-diffusion type with delay and which are nonlinear and associated with variable coefficients. A novel technique is used in this study to obtain the exact solutions which are new and are of the form of traveling-wave solutions. Arbitrary functions are present in the solutions and they also contain free parameters, which make them suitable for usage in solving certain modeling problems, testing numerical and approximate analytical methods. The results of this study also find applications in obtaining the exact solutions of other forms of partial differential equations which are more complex. Specific examples of nonlinear equations of reaction-diffusion type with delay are given and their exact solutions are presented. Solutions of certain reaction-diffusion equations are also displayed graphically.

... Interested readers in the exact solutions of (2.12) for varieties of the functions a(x), p(x) and q(x), are referred to [24,25]. ...

... , для которых уравнение (2) допускает решение в замкнутой форме [18]. Поэтому если в какой-нибудь статье будет доказана интегрируемость этого уравнения для других значений a, этот результат безусловно будет новым. ...

The problems of estimating the similarity index of inhomogeneous scientific publications containing equations and formulas are discussed for the first time. It is shown that the presence of equations and formulas (as well as figures, drawings, and tables) is a complicating factor that significantly complicates the study of such texts. It has been proved that the method for determining the similarity index of publications, based on taking into account individual mathematical symbols and parts of equations and formulas, is ineffective and can lead to erroneous and even completely absurd conclusions. Possibilities of the most popular software systems Antiplagiat and iThenticate, currently used in scientific journals, are investigated for detecting plagiarism and self-plagiarism. The results of processing by the iThenticate system of specific examples and specific test problems containing equations and formulas are presented. It has been established that this software system, when analyzing heterogeneous texts, is often unable to distinguish self-plagiarism from pseudo-self-plagiarism, seeming real (but false and imaginary) self-plagiarism. A model complex situation is considered, in which the identification of self-plagiarism requires the involvement of highly qualified specialists of a narrow profile. Various ways to improve the work of software systems for comparing inhomogeneous texts are proposed. This article will be useful to researchers and university teachers in physics, mathematics, and engineering, programmers dealing with problems in image recognition and research topics of digital image processing, as well as a wide range of readers who are interested in issues of plagiarism and self-plagiarism.

Abstract in English. The problems of estimating the similarity index of inhomogeneous scientific publications containing equations and formulas are discussed for the first time. It is shown that the presence of equations and formulas (as well as figures, drawings, and tables) is a complicating factor that significantly complicates the study of such texts. It has been proved that the method for determining the similarity index of publications, based on taking into account individual mathematical symbols and parts of equations and formulas, is ineffective and can lead to erroneous and even completely absurd conclusions. Possibilities of the most popular software systems Antiplagiat and iThenticate, currently used in scientific journals, are investigated for detecting plagiarism and self-plagiarism. The results of processing by the iThenticate system of specific examples and specific test problems containing equations and formulas are presented. It has been established that this software system, when analyzing heterogeneous texts, is often unable to distinguish self-plagiarism from pseudo-self-plagiarism, seeming real (but false and imaginary) self-plagiarism. A model complex situation is considered, in which the identification of self-plagiarism requires the involvement of highly qualified specialists of a narrow profile. Various ways to improve the work of software systems for comparing inhomogeneous texts are proposed. This article will be useful to researchers and university teachers in physics, mathematics, and engineering, programmers dealing with problems in image recognition and research topics of digital image processing, as well as a wide range of readers who are interested in issues of plagiarism and self-plagiarism. ******* Аннотация на русском языке. Впервые обсуждаются проблемы оценки индекса подобия неоднородных научных публикаций, содержащих уравнения и формулы. Показано, что наличие уравнений и формул (а также графиков, рисунков и таблиц) является осложняющим фактором, существенно затрудняющим исследование таких текстов. Доказано, что метод определения индекса подобия публикаций, основанный на учете отдельных математических символов и частей уравнений и формул, является неэффективным и может приводить к ошибочным и даже совершенно абсурдным выводам. Исследуются возможности наиболее популярных программных систем Антиплагиат и iThenticate, используемых в настоящее время в научных журналах для выявления плагиата и самоплагиата. Приведены результаты обработки системой iThenticate конкретных примеров и специальных тестовых задач, содержащих уравнения и формулы. Установлено, что эта программная система при анализе неоднородных текстов часто неспособна отличить самоплагиат от псевдосамоплагиата -- кажущегося (ложного, мнимого) самоплагиата. Рассмотрена модельная сложная ситуация, в которой идентификация самоплагиата требует привлечения высококвалифицированных специалистов узкого профиля. Предлагаются различные пути улучшения работы программных систем для сопоставления неоднородных текстов. Данная статья будет полезна научным работникам и преподавателям вузов физико-математического и инженерного профиля, программистам, занимающимся проблемой распознавания образов и вопросами цифровой обработки изображений, а также широкому кругу читателей, которые интересуются вопросами плагиата и самоплагиата.

... Notably, the overwhelming majority of known general solutions to nonlinear ODEs are presented in an implicit or parametric form (a similar conclusion follows from the statistical processing of most comprehensive reference books on exact solutions of ODEs [273,276]). This circumstance allows us to state a plausible hypothesis that nonlinear PDEs also admit exact solutions (by quadrature) in an implicit or parametric form more often than in explicit form. ...

This book is devoted to describing and applying methods of generalized and functional separation of variables used to find exact solutions of nonlinear partial differential equations (PDEs). It also presents the direct method of symmetry reductions and its more general version. In addition, the authors describe the differential constraint method, which generalizes many other exact methods. The presentation involves numerous examples of utilizing the methods to find exact solutions to specific nonlinear equations of mathematical physics. The equations of heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion theory, chemical technology, biology, and other disciplines are studied. Particular attention is paid to nonlinear equations of a reasonably general form that depend on one or several arbitrary functions. Such equations are the most difficult to analyze. Their exact solutions are of significant practical interest, as they are suitable to assess the accuracy of various approximate analytical and numerical methods. The book contains new material previously unpublished in monographs. It is intended for a broad audience of scientists, engineers, instructors, and students specializing in applied and computational mathematics, theoretical physics, mechanics, control theory, chemical engineering science, and other disciplines. Individual sections of the book and examples are suitable for lecture courses on partial differential equations, equations of mathematical physics, and methods of mathematical physics, for delivering special courses and for practical training.

... The theory of ordinary differential equations establishes that eq. (10) has a unique solution in the open interval (0, ) if functions ( ) and ( ) are continuous in the domain ∈ (0, ) (p. 96 [25]), which is the case as long as ( ) is continuous in the domain ∈ (0, ). We note that the singularity of ( ) at = 0 does not belong to the open interval (0, ). ...

Sergio Barbero

Multifocal lenses comprising progressive power surfaces are commonly used in contact and intraocular lens designs. Given a visual performance metric, a wavefront engineering approach to design such lenses is based on searching for the optimal wavefront at the exit pupil of the eye. Multifocal wavefronts distribute the energy along the different foci thanks to having a varying mean curvature. Therefore, a fundamental step in the wavefront engineering approach is to generate the wavefront from a prescribed mean curvature function. Conventionally, such a thing is done by superimposing spherical wavefront patches and maybe adding a certain component of spherical aberration to each spherical patch in order to increase the depth-of-field associated with each focus. However, such a procedure does not lead to smooth wavefront solutions and also restricts the type of available multifocal wavefronts. We derive a new, to the best of our knowledge, mathematical method to uniquely construct multifocal wavefronts from mean curvature functions (depending on radial and angular coordinates) under certain numerically justified approximations and restrictions. Additionally, our procedure leads to a particular family of wavefronts (line-umbilical multifocal wavefronts) described by 2 conditions: (1) to be smooth multiplicative separable functions in the radial and angular coordinates; (2) to be umbilical along a specific segment connecting the circle center with its edge. We provide several examples of multifocal wavefronts belonging to this family, including a smooth variant of the so-called light sword element.

Igoris Belovas

The paper extends the research on the series with binomial-like coefficients for the computation of zeta functions on the complex plane. It offers alternative perspectives on the proof of central limit theorems for the coefficients of the series. The moment generating function of the coefficients and exact expressions for the first moments of the coefficients of the series are established.

Roman Cherniha
Vasyl' Davydovych

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

Richard Banach
Huibiao Zhu

In view of the increasing importance of cyber-physical systems, and of their correct design, the Abstract State Machine (ASM) framework is extended to include continuously varying quantities as well as the conventional discretely changing ones. This opens the door to the more faithful modelling of many scenarios where digital systems have to interact with the continuously varying physical world. Transitions in the extended framework are thus either moded (catering for discontinuously changing quantities), or pliant (catering for smoothly changing quantities). An operational semantics is provided, first for monolithic systems, and this is then extended to give a semantics for systems consisting of several distinct subsystems. This allows each subsystem to undergo its own subsystem-specific mode and pliant transitions. Refinement is elaborated in the extended context for both monolithic and composed systems. The formalism is illustrated using an example of a bouncing tennis ball.

The method of non-local transformations is proposed for numerical integration of non-linear Cauchy problems having non-monotonic blow-up solutions. In such problems there exists a singular point whose position is unknown in advance (for this reason, the standard numerical methods for solving blow-up problems can lead to significant errors). In addition, the non-monotonic behavior of the solution excludes a possibility of applying the hodograph transformation and some other methods that are used for numerical investigation of simpler problems having monotone blow-up solutions. In this paper, the method is described for numerical integration of similar problems for non-linear n th-order ordinary differential equations x_t^((n))=f(t,x,x_t^',…,x_t^((n-1) ) ), based on the introduction of a new non-local independent variable ξ, which is related to the original variables t and x by the equation ξ_t^'=g(t,x,x_t^',…,x_t^((n-1) ),ξ), and the subsequent transformation of the original problem to the Cauchy problem for the corresponding system of first-order differential equations. With a suitable choice of the regularizing function g, the proposed method leads to problems whose solutions are presented in parametric form and do not have blowing-up singular points; therefore the transformed problems allow the application of the standard fixed-step numerical methods. A number of test problems with non-monotonic blow-up is constructed for differential equations of the first, second, third, and fourth orders, which have exact solutions expressed in elementary functions. The numerical integration of test problems has shown high efficiency of methods based on non-local transformations of a special type (and the practical inapplicability of the arc-length transformation for numerical solving blow-up problems with ODEs of high order). In addition to problems for single ODEs, Cauchy problems for systems of coupled equations also are considered. Some recommendations are given on a suitable choice of a variable step in direct adaptive numerical methods in which the transformations of equations are not used. It is important to note that the method of non-local transformations can also be used to numerically integrate other problems with large solution gradients (including problems of boundary-layer type).

We consider blow-up problems having non-monotonic singular solutions that tend to infinity at a previously unknown point. For second-, third-, and fourth-order nonlinear ordinary differential equations, the corresponding multi-parameter test problems allowing exact solutions in elementary functions are proposed for the first time. A method of non-local transformations, that allows to numerically integrate non-monotonic blow-up problems, is described. A comparison of exact and numerical solutions showed the high efficiency of this method. It is important to note that the method of non-local transformations can be useful for numerical integration of other problems with large solution gradients (for example, in problems with solutions of boundary-layer type).

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions are unknown a priori (the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative. The second method is based on introducing a new auxiliary non-local variable with the subsequent transformation to the Cauchy problem for the corresponding system of ODEs. The third method is based on adding to the original equation of a differential constraint, which is an auxiliary ODE connecting the given variables and a new variable. The proposed methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the transformed problems admit the application of standard fixed-step numerical methods. The efficiency of these methods is illustrated by solving a number of test problems that admit an exact analytical solution. It is shown that: (i) the methods based on non-local transformations of a special kind are more efficient than several other methods, (ii) among the proposed methods, the most general method is the method based on the differential constraints. Some examples of nonclassical blow-up problems are considered. Simple theoretical estimates are derived for the critical value of an independent variable. It is shown that the method based on a non-local transformation of the general form as well as the method based on the differential constraints admit generalizations to the nth-order ODEs and systems of coupled ODEs.

Two new methods of numerical integration of Cauchy problems for nonlinear ODEs of the first- and second-order, which have blow-up solutions are described. In such problems, the position of the singular point is not known in advance. The first method is based on obtaining an equivalent system of equations by applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable, t=yx′. The second method is based on introducing a new auxiliary non-local variable of the form ξ=∫x0xg(x,y,yx′)dx with the subsequent transformation to the Cauchy problem for the corresponding system of coupled ODEs. Both methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the standard fixed-step numerical methods can be applied. The efficiency of the proposed methods is illustrated with a number of test problems that admit exact solutions. It is shown that the methods, based on special exp-type transformations (which are particular cases of the general non-local transformation), are more efficient than the method based on the hodograph transformation, the method of the arc-length transformation, and the method based on the differential transformation. The method, based on introducing a non-local variable, can be generalized to the n th-order ODEs and systems of coupled ODEs. ***** This article is available on the author's homepage, see http://eqworld.ipmnet.ru/Arts_Polyanin/NLM_2017_Polyanin_Shingareva.pdf

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

The study deals with nonlinear ordinary differential equations defined parametrically by two relations; these arise in fluid dynamics and are a special class of coupled differential-algebraic equations. We propose a few techniques for reducing such equations, first or second order, to systems of standard ordinary differential equations as well as techniques for the exact integration of these systems. Several examples show how to construct general solutions to some classes of nonlinear equations involving arbitrary functions. We specify a procedure for the numerical solution of the Cauchy problem for parametrically defined differential equations and related differential-algebraic equations. The proposed techniques are also effective for the numerical integration of problems for implicitly defined equations. ***** This article is available on the author's homepage, see http://eqworld.ipmnet.ru/Arts_Polyanin/AML_2017_Polyanin_Zhurov.pdf