Polyanin Zaitsev Handbook of ODEs 2017

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/320776882
Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and
Problems
Book · January 2018
DOI: 10.1201/9781315117638
CITATIONS
READS
117
142,975
2 authors:
Andrei D. Polyanin
Valentin Zaitsev
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences
Herzen State Pedagogical University of Russia
452 PUBLICATIONS 9,799 CITATIONS
38 PUBLICATIONS 3,862 CITATIONS
SEE PROFILE
All content following this page was uploaded by Andrei D. Polyanin on 01 November 2017.
The user has requested enhancement of the downloaded file.
SEE PROFILE
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 differential equations with solutions, as well as exact, asymptotic, approximate analytical, numerical, 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 solutions, 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. Symbolic 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 essential 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, hypergeometric, 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 qualitative features or singularities.
• Symbolic and numerical methods with Maple, Mathematica, and MATLAB.
xxxiii
xxxiv
P REFACE
• 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 differential 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 topics 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 students. 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 LizárragaCelaya, 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 researchers, university and college teachers, engineers, and students in various fields of mathematics, physics, mechanics, control, chemistry, economics, and engineering sciences.
Andrei D. Polyanin
Valentin F. Zaitsev
CONTENTS
Preface
xxiii
Authors
xxv
Basic Notation and Remarks
Part I. Methods for Ordinary Differential Equations
1 Methods for First-Order Differential Equations
1.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . .
1.1.1 Equations Solved for the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Equations Not Solved for the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Equations Solved for the Derivative. Simplest Techniques of Integration . . . . . .
1.2.1 Equations with Separable Variables and Related Equations . . . . . . . . . . . .
1.2.2 Homogeneous and Generalized Homogeneous Equations . . . . . . . . . . . . .
1.2.3 Linear Equation and Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Darboux Equation and Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Exact Differential Equations. Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 General Riccati Equation. Simplest Integrable Cases. Polynomial
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Use of Particular Solutions to Construct the General Solution . . . . . . . . .
1.4.3 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Special Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Abel Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 General Form of Abel Equations of the First Kind. Simplest Integrable
Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 General Form of Abel Equations of the Second Kind. Simplest Integrable
Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.3 Use of Particular Solutions to Construct Self-Transformations and the
General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Classification and Specific Features of Some Classes of Solutions . . . . . . . . . . . .
1.7.1 Stable and Unstable Solutions. Equilibrium Points . . . . . . . . . . . . . . . . . . .
1.7.2 Blow-Up Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.3 Space Localization of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.4 Cauchy Problems Admitting Non-Unique Solutions . . . . . . . . . . . . . . . . . .
1.8 Equations Not Solved for the Derivative and Equations Defined Parametrically
1.8.1 Method of “Integration by Differentiation” for Equations Not Solved for
the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.2 Equations Not Solved for the Derivative. Specific Equations . . . . . . . . . .
1.8.3 Equations Defined Parametrically and Differential-Algebraic Equations
xxvii
1
3
3
3
7
8
8
9
10
11
12
12
13
13
13
15
16
17
18
18
19
20
20
21
22
25
25
28
33
36
37
37
38
40
1.9 Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9.1 General Form of Contact Transformations. Method for the Construction
of Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9.2 Examples of Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Pfaffian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.2 Completely Integrable Pfaffian Equations . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.3 Pfaffian Equations Not Satisfying the Integrability Condition . . . . . . . .
1.11 Approximate Analytic Methods for Solution of ODEs . . . . . . . . . . . . . . . . . . . . .
1.11.1 Method of Successive Approximations (Picard Method) . . . . . . . . . . . .
1.11.2 Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11.3 Method of Series Expansion in the Independent Variable . . . . . . . . . . .
1.11.4 Method of Regular Expansion in the Small Parameter . . . . . . . . . . . . . .
1.12 Differential Inequalities and Solution Estimates . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12.1 Two Theorems on Solution Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12.2 Chaplygin’s Theorem and Its Applications (Bilateral Estimates of the
Cauchy Problem Solution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13 Standard Numerical Methods for Solving Ordinary Differential Equations . . .
1.13.1 Single-Step Methods. Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . .
1.13.2 Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13.3 Predictor–Corrector Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13.4 Modified Multistep Methods (Butcher’s Methods) . . . . . . . . . . . . . . . . .
1.13.5 Stability and Convergence of Numerical Methods . . . . . . . . . . . . . . . . . .
1.13.6 Well- and Ill-Conditioned Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14 Special Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14.1 Special Methods Based on Auxiliary Equations . . . . . . . . . . . . . . . . . . .
1.14.2 Numerical Integration of Equations That Contain Fixed Singular
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14.3 Numerical Integration of Equations Defined Parametrically or
Implicitly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14.4 Numerical Solution of Blow-Up Problems . . . . . . . . . . . . . . . . . . . . . . . .
1.14.5 Numerical Solution of Problems with Root Singularity . . . . . . . . . . . . .
42
42
43
44
44
45
48
49
49
50
53
56
57
57
58
61
61
68
70
72
72
73
74
74
76
79
81
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 yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.4.2 Equations Containing yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.5 Boundary Value Problems. Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 First, Second, Third, and Some Other Boundary Value Problems . . . . . . .
2.5.2 Simplification of Boundary Conditions. Self-Adjoint Form of Equations
2.5.3 Green’s and Modified Green’s Functions. Representation Solutions via
Green’s or Modified Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Sturm–Liouville Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 General Properties of the Sturm–Liouville Problem (2.6.1.1), (2.6.1.2) .
2.6.3 Problems with Boundary Conditions of the First Kind . . . . . . . . . . . . . . . .
2.6.4 Problems with Boundary Conditions of the Second Kind . . . . . . . . . . . . .
2.6.5 Problems with Boundary Conditions of the Third Kind . . . . . . . . . . . . . . .
2.6.6 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Theorems on Estimates and Zeros of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Theorem on Estimates of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Sturm Comparison Theorem on Zeros of Solutions . . . . . . . . . . . . . . . . . .
2.7.3 Qualitative Behavior of Solutions as x → ∞ . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 Numerov’s Method (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.2 Modified Shooting Method (Boundary Value Problems) . . . . . . . . . . . . . .
2.8.3 Sweep Method (Boundary Value Problems) . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.4 Method of Accelerated Convergence in Eigenvalue Problems . . . . . . . . .
2.8.5 Well-Conditioned and Ill-Conditioned Problems . . . . . . . . . . . . . . . . . . . . .
3 Methods for Second-Order Nonlinear Differential Equations
3.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . .
3.1.1 Equations Solved for the Derivative. General Solution . . . . . . . . . . . . . . . .
3.1.2 Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . .
3.2 Some Transformations. Equations Admitting Reduction of Order . . . . . . . . . . . .
3.2.1 Equations Not Containing y or x Explicitly. Related Equations . . . . . . . .
3.2.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Generalized Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Equations Invariant under Scaling–Translation Transformations . . . . . . .
3.2.5 Exact Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Nonlinear Equations Involving Linear Homogeneous Differential Forms
3.2.7 Reduction of Quasilinear Equations to the Normal Form . . . . . . . . . . . . . .
3.2.8 Equations Defined Parametrically and Differential-Algebraic Equations
3.3 Boundary Value Problems. Uniqueness and Existence Theorems. Nonexistence
Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Uniqueness and Existence Theorems for Boundary Value Problems . . . .
3.3.2 Reduction of Boundary Value Problems to Integral Equations. Integral
Identity. Jentzch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Theorem on Nonexistence of Solutions to the First Boundary Value
Problem. Theorems on Existence of Two Solutions . . . . . . . . . . . . . . . . . .
3.3.4 Examples of Existence, Nonuniqueness, and Nonexistence of Solutions
to First Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Theorems on Nonexistence of Solutions for the Mixed Problem.
Theorems on Existence of Two Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.6 Examples of Existence, Nonuniqueness, and Nonexistence of Solutions
to Mixed Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.7 Theorems on Existence of Two Solutions for the Third Boundary Value
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
103
106
107
110
110
111
112
113
114
114
115
115
115
116
116
116
117
118
119
120
123
123
123
123
124
124
125
126
126
127
128
129
129
133
134
137
139
142
145
148
151
3.4
3.5
3.6
3.7
3.8
3.3.8 Boundary Value Problems for Linear Equations with Nonlinear
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Method of Regular Series Expansions with Respect to the Independent Variable
3.4.1 Method of Expansion in Powers of the Independent Variable . . . . . . . . . .
3.4.2 Padé Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Movable Singularities of Solutions of Ordinary Differential Equations. Painlevé
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Preliminary Remarks. Singular Points of Solutions . . . . . . . . . . . . . . . . . .
3.5.2 First Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Second Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.4 Third Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 Fourth Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.6 Fifth Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.7 Sixth Painlevé Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Preliminary Remarks. Summary Table of Basic Methods . . . . . . . . . . . . .
3.6.2 Method of Regular (Direct) Expansion in Powers of the Small Parameter
3.6.3 Method of Scaled Parameters (Lindstedt–Poincaré Method) . . . . . . . . . . .
3.6.4 Averaging Method (Van der Pol–Krylov–Bogolyubov Scheme) . . . . . . . .
3.6.5 Method of Two-Scale Expansions (Cole–Kevorkian Scheme) . . . . . . . . .
3.6.6 Method of Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . .
Galerkin Method and Its Modifications (Projection Methods) . . . . . . . . . . . . . . . .
3.7.1 Approximate Solution for a Boundary Value Problem . . . . . . . . . . . . . . . .
3.7.2 Galerkin Method. General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.3 Bubnov–Galerkin, Moment, and Least Squares Methods . . . . . . . . . . . . . .
3.7.4 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.5 Method of Partitioning the Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.6 Least Squared Error Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Method of Successive Approximations (Cauchy Problem) . . . . . . . . . . . .
3.8.2 Runge–Kutta Method (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.3 Reduction to a System of Equations (Cauchy Problem) . . . . . . . . . . . . . . .
3.8.4 Predictor–Corrector Methods (Cauchy Problem) . . . . . . . . . . . . . . . . . . . . .
3.8.5 Shooting Method (Boundary Value Problems) . . . . . . . . . . . . . . . . . . . . . . .
3.8.6 Numerical Methods for Problems with Equations Defined Implicitly or
Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.7 Numerical Solution Blow-Up Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Methods for Linear ODEs of Arbitrary Order
4.1 Linear Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Homogeneous Linear Equations. General Solution . . . . . . . . . . . . . . . . . . .
4.1.2 Nonhomogeneous Linear Equations. General and Particular Solutions . .
4.2 Linear Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Homogeneous Linear Equations. General Solution. Order Reduction.
Liouville Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Nonhomogeneous Linear Equations. General Solution. Superposition
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Nonhomogeneous Linear Equations. Cauchy Problem. Reduction to
Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
154
154
156
157
157
158
160
162
165
167
169
171
171
173
174
175
177
178
181
181
182
182
183
184
185
185
185
186
186
186
187
190
192
197
197
197
198
200
200
201
202
4.3 Laplace Transform and the Laplace Integral. Applications to Linear ODEs . . . .
4.3.1 Laplace Transform and the Inverse Laplace Transform . . . . . . . . . . . . . . .
4.3.2 Main Properties of the Laplace Transform. Inversion Formulas for Some
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Limit Theorems. Representation of Inverse Transforms as Convergent
Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Solution of the Cauchy Problem for Constant-Coefficient Linear ODEs.
Applications to Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Solution of Linear Equations with Polynomial Coefficients Using the
Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.6 Solution of Linear Equations with Polynomial Coefficients Using the
Laplace Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Asymptotic Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Fourth-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Higher-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Statement of the Problem. Approximate Solution . . . . . . . . . . . . . . . . . . . .
4.5.2 Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204
204
205
207
209
211
212
213
213
214
215
215
215
5 Methods for Nonlinear ODEs of Arbitrary Order
5.1 General Concepts. Cauchy Problem. Uniqueness and Existence Theorems . . . .
5.1.1 Equations Solved for the Derivative. General Solution . . . . . . . . . . . . . . . .
5.1.2 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Equations Not Containing y or x Explicitly . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Generalized Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Equations Invariant under Scaling-Translation Transformations . . . . . . . .
5.2.5 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Method for Construction of Solvable Equations of General Form . . . . . . . . . . . .
5.3.1 Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Numerical Integration of n-order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Numerical Solution of the Cauchy Problem for n-order ODEs . . . . . . . . .
5.4.2 Numerical Solution of Equations Defined Implicitly or Parametrically . .
217
217
217
218
218
218
219
220
220
220
222
222
223
224
224
224
6 Methods for Linear Systems of ODEs
6.1 Systems of Linear Constant-Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Systems of First-Order Linear Homogeneous Equations. General
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Systems of First-Order Linear Homogeneous Equations. Particular
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3 Nonhomogeneous Systems of Linear First-Order Equations . . . . . . . . . . .
6.1.4 Homogeneous Linear Systems of Higher-Order Differential Equations .
6.1.5 Normal Coordinates and Natural Oscillations . . . . . . . . . . . . . . . . . . . . . . .
6.1.6 Nonhomogeneous Higher-Order Linear Systems. D’Alembert’s Method
6.1.7 Usage of the Laplace Transform for Solving Linear Systems of
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.8 Classification of Equilibrium Points of Two-Dimensional Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
227
227
227
228
230
231
232
233
234
235
6.2 Systems of Linear Variable-Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Homogeneous Systems of Linear First-Order Equations . . . . . . . . . . . . . .
6.2.2 Nonhomogeneous Systems of Linear First-Order Equations . . . . . . . . . . .
6.2.3 Euler System of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . .
240
240
242
243
7 Methods for Nonlinear Systems of ODEs
7.1 Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . . .
7.1.1 Systems Solved for the Derivative. A Solution and the General Solution
7.1.2 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Reduction of Systems of Equations to a Single Equation or to an
Autonomous System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.4 First Integrals. Using Them to Reduce System Dimension . . . . . . . . . . . .
7.2 Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . . .
7.2.1 Integrable Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Autonomous Systems and Their Reduction to Systems of Lower
Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Lyapunov Stability. Asymptotic Stability. Unstable Solutions . . . . . . . . .
7.3.2 Theorems of Stability and Instability by First Approximation . . . . . . . . .
7.3.3 Lyapunov Function. Theorems of Stability and Instability . . . . . . . . . . . .
7.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Systems Involving Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . .
245
245
245
245
8 Elements of Bifurcation Theory
8.1 Dynamical Systems. Rough and Nonrough Systems . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Bifurcation. Dynamical Systems. Phase Portrait . . . . . . . . . . . . . . . . . . . . .
8.1.2 Topologically Equivalent Systems. Rough and Nonrough Systems . . . . .
8.2 Bifurcations of Second-Order Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Second-Order Dynamical Systems. Rough and Nonrough Systems . . . . .
8.2.2 Bifurcations in Systems of the First Degree of Nonroughness . . . . . . . . .
8.3 Bifurcations of Solutions to Boundary Value Problems . . . . . . . . . . . . . . . . . . . . .
8.3.1 Bifurcations of Solutions to Linear Boundary Value Problems . . . . . . . . .
8.3.2 Bifurcations in Solutions to Nonlinear Boundary Value Problems . . . . . .
8.3.3 Bifurcation Analysis of Boundary Value Problems without Linearizing
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Elementary Theory of Using Invariants for Solving Equations
9.1 Introduction. Symmetries. General Scheme of Using Invariants for Solving
Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Symmetries. Transformations Preserving the Form of Equations.
Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2 General Scheme of Using Invariants for Solving Mathematical
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Algebraic Equations and Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Algebraic Equations with Even Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Reciprocal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Systems of Algebraic Equations Symmetric with Respect to Permutation
of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
246
247
248
248
249
250
250
251
253
255
255
259
263
263
263
264
265
265
266
270
270
270
273
277
277
277
279
279
279
280
282
9.3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Transformations Preserving the Form of Equations. Invariants . . . . . . . . .
9.3.2 Order Reduction Procedure for Equations with n ≥ 2 (Reduction to
Solvable Form with n = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 Simple Transformations. Invariant Determination Procedure . . . . . . . . . .
9.3.4 Analysis of Some Ordinary Differential Equations. Useful Remarks . . . .
10 Methods for the Construction of Particular Solutions
10.1 Two Problems on Searching for Particular Solutions to ODEs with Parameters
10.1.1 Preliminary Remarks. Traveling Wave Solutions . . . . . . . . . . . . . . . . . .
10.1.2 Two Problems for ODEs with Parameters. Conditional Capacity of
Exact Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Method of Undetermined Coefficients and Its Special Cases . . . . . . . . . . . . . . . .
10.2.1 General Description of the Method of Undetermined Coefficients . . . .
10.2.2 Power-Law, Tanh-Coth, and Sine-Cosine Methods . . . . . . . . . . . . . . . . .
10.2.3 Exp-Function, Q-Expansion and Related Methods . . . . . . . . . . . . . . . . .
10.3 Method of Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Preliminary Remarks. First-Order Differential Constraints and Their
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Differential Constraints of Arbitrary Order. General Consistency
Method for Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3 Using Point Transformations in Combination with the Method of
Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4 Using Several Differential Constraints. G′/G-Expansion Method and
Simplest Equation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Group Methods for ODEs
11.1 Lie Group Method. Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1 Local One-Parameter Lie Group of Transformations. Invariance
Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.2 Group Analysis of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . .
11.1.3 Utilization of Local Groups for Reducing the Order of Equations and
Their Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.4 Seeking Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Contact and Bäcklund Transformations. Formal Operators. Factorization
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Contact Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Bäcklund Transformations. Formal Operators and Nonlocal Variables
11.2.3 Factorization Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 First Integrals (Conservation Laws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Algorithm of Finding First Integrals of ODEs . . . . . . . . . . . . . . . . . . . . .
11.3.2 Applications to Second-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 Lie–Bäcklund Symmetries Generated by First Integrals . . . . . . . . . . . . .
11.4 Underdetermined Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.2 Factorization Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.3 Some Technical Elements. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.4 On Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
283
283
284
284
285
289
289
289
290
292
292
293
297
302
302
306
307
311
313
313
313
316
319
321
322
322
323
326
334
334
336
337
345
345
346
348
350
12 Discrete-Group Methods
12.1 Discrete Group Method for Point Transformations . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Classes of ODEs with Parameters. Discrete Group of Point
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Discrete Group Method Based on RF-Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 General Description of the Method. First and Second RF-Pairs . . . . . .
12.2.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Discrete Group Method Based on the Inclusion Method . . . . . . . . . . . . . . . . . . .
355
355
355
355
358
358
359
364
Part II. Exact Solutions of Ordinary Differential Equations
365
13 First-Order Ordinary Differential Equations
13.1 Simplest Equations with Arbitrary Functions Integrable in Closed Form . . . . .
13.1.1 Equations of the Form yx′ = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.2 Equations of the Form yx′ = f (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.3 Separable Equations yx′ = f (x)g(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.4 Linear Equation g(x)yx′ = f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . . . .
13.1.5 Bernoulli Equation g(x)yx′ = f1 (x)y + fn (x)y n . . . . . . . . . . . . . . . . . . .
13.1.6 Homogeneous Equation yx′ = f (y/x) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Riccati Equation g(x)yx′ = f2 (x)y 2 + f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
13.2.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
13.2.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
13.2.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
13.2.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . .
13.2.8 Equations with Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.9 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Equations of the Form yyx′ − y = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.2 Equations of the Form yyx′ = f (x)y + 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.3 Equations of the Form yyx′ = f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . .
13.3.4 Equations of the Form [g1 (x)y + g0 (x)]yx′ = f2 (x)y 2 + f1 (x)y + f0 (x)
13.3.5 Some Types of First- and Second-Order Equations Reducible to Abel
Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Equations Containing Polynomial Functions of y . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Abel Equations of the First Kind
yx′ = f3 (x)y 3 + f2 (x)y 2 + f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.2 Equations of the Form (A22 y 2 + A12 xy + A11 x2 + A0 )yx′ =
B22 y 2 + B12 xy + B11 x2 + B0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.3 Equations of the Form (A22 y 2 + A12 xy + A11 x2 + A2 y + A1 x)yx′ =
B22 y 2 + B12 xy + B11 x2 + B2 y + B1 x . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.4 Equations of the Form (A22 y 2 +A12 xy+A11 x2 +A2 y+A1 x+A0 )yx′ =
B22 y 2 + B12 xy + B11 x2 + B2 y + B1 x + B0 . . . . . . . . . . . . . . . . . . . . . . .
13.4.5 Equations of the Form (A3 y 3 +A2 xy 2 +A1 x2 y+A0 x3 +a1 y+a0 x)yx′ =
B3 y 3 + B2 xy 2 + B1 x2 y + B0 x3 + b1 y + b0 x . . . . . . . . . . . . . . . . . . . . . .
367
367
367
367
367
368
368
368
368
368
369
375
378
380
382
387
389
392
394
394
409
410
422
429
431
431
436
438
447
452
13.5 Equations of the Form f (x, y)yx′ = g(x, y) Containing Arbitrary Parameters .
13.5.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
13.5.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
13.5.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
13.5.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
13.5.6 Equations Containing Combinations of Exponential, Hyperbolic,
Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .
13.6 Equations of the Form F (x, y, yx′ ) = 0 Containing Arbitrary Parameters . . . . .
13.6.1 Equations of the Second Degree in yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6.2 Equations of the Third Degree in yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6.3 Equations of the Form (yx′ )k = f (y) + g(x) . . . . . . . . . . . . . . . . . . . . . . .
13.6.4 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.7 Equations of the Form f (x, y)yx′ = g(x, y) Containing Arbitrary Functions . .
13.7.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.7.2 Equations Containing Exponential and Hyperbolic Functions . . . . . . .
13.7.3 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
13.7.4 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
13.7.5 Equations Containing Combinations of Exponential, Logarithmic, and
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8 Equations Not Solved for the Derivative and Equations Defined Parametrically
13.8.1 Equations Not Solved for the Derivative Containing Arbitrary
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.2 Some Transformations of Equations Not Solved for the Derivative . . .
13.8.3 Equations Defined Parametrically Containing Arbitrary Functions . . .
14 Second-Order Ordinary Differential Equations
14.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.1 Representation of the General Solution through a Particular Solution .
14.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
14.1.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
14.1.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
14.1.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
14.1.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . .
14.1.8 Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.9 Equations with Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.10 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ = F (y, y ′ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Autonomous Equations yxx
x
′′ − y ′ = f (y) . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 Equations of the Form yxx
x
′′ + f (y)y ′ + y = 0 . . . . . . . . . . . . . . . . . . . . . .
14.2.2 Equations of the Form yxx
x
′′ + f (y)y ′ + g(y) = 0 . . . . . . . . . . . . . . . . . . . . . . .
14.2.3 Lienard Equations yxx
x
′′ + f (y ′ ) + g(y) = 0 . . . . . . . . . . . . . . . . . . . . . . .
14.2.4 Rayleigh Equations yxx
x
′′
14.3 Emden–Fowler Equation yxx = Axn y m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.2 First Integrals (Conservation Laws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.3 Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
456
456
460
464
466
467
469
471
471
478
481
490
495
495
498
500
501
502
504
504
514
515
519
519
519
519
553
560
565
568
580
586
595
603
605
606
610
613
616
619
619
625
628
′′ = A xn1 y m1 + A xn2 y m2 . . . . . . . . . . . . . . . . . . . . .
14.4 Equations of the Form yxx
1
2
14.4.1 Classification Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4.2 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ = Axn y m (y ′ )l . . . . . . . . . . . . . . . . . .
14.5 Generalized Emden–Fowler Equation yxx
x
14.5.1 Classification Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5.2 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5.3 Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ = A xn1 y m1 (y ′ )l1 + A xn2 y m2 (y ′ )l2 . . . . . . . . . . .
14.6 Equations of the Form yxx
1
2
x
x
′′ = A x−1 y ′ + A xn y m . . . . . .
14.6.1 Modified Emden–Fowler Equation yxx
1
2
x
′′ = (A xn1 y m1 + A xn2 y m2 )(y ′ )l . . . . . . . .
14.6.2 Equations of the Form yxx
1
2
x
′′ = σAxn y m (y ′ )l + Axn−1 y m+1 (y ′ )l−1 . . .
14.6.3 Equations of the Form yxx
x
x
14.6.4 Other Equations (l1 ̸= l2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ = f (x)g(y)h(y ′ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7 Equations of the Form yxx
x
′′ = f (x)g(y) . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7.1 Equations of the Form yxx
14.7.2 Equations Containing Power Functions (h ̸≡ const) . . . . . . . . . . . . . . . .
14.7.3 Equations Containing Exponential Functions (h ̸≡ const) . . . . . . . . . . .
14.7.4 Equations Containing Hyperbolic Functions (h ̸≡ const) . . . . . . . . . . . .
14.7.5 Equations Containing Trigonometric Functions (h ̸≡ const) . . . . . . . . .
14.7.6 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.8 Some Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . .
14.8.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.8.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
14.8.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
14.8.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
14.8.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
14.8.6 Equations Containing the Combinations of Exponential, Hyperbolic,
Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .
14.9 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ + G(x, y) = 0 . . . . . . . . . . . . . . . . . .
14.9.1 Equations of the Form F (x, y)yxx
′′ + G(x, y)y ′ + H(x, y) = 0 . . . . . . .
14.9.2 Equations of the Form F (x, y)yxx
x
∑M
′′ +
′ m
14.9.3 Equations of the Form F (x, y)yxx
G
m=0 m (x, y)(yx ) = 0
(M = 2, 3, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′ + G(x, y, y ′ ) = 0 . . . . . . . . . . . .
14.9.4 Equations of the Form F (x, y, yx′ )yxx
x
14.9.5 Equations Not Solved for Second Derivative . . . . . . . . . . . . . . . . . . . . . .
14.9.6 Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.9.7 Equations Defined Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.9.8 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 Third-Order Ordinary Differential Equations
15.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.3 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
15.1.4 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
15.1.5 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
15.1.6 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
15.1.7 Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . .
15.1.8 Equations Containing Combinations of Exponential, Logarithmic,
Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.9 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .
628
628
632
652
652
657
676
678
678
688
718
733
739
739
742
746
750
751
752
753
753
761
769
775
777
783
786
786
793
798
802
813
815
822
825
829
829
829
830
848
853
863
866
879
884
892
′′′ = Axα y β (y ′ )γ (y ′′ )δ . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Equations of the Form yxxx
xx
x
15.2.1 Classification Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′′ = Ay β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2.2 Equations of the Form yxxx
′′′ = Axα y β . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2.3 Equations of the Form yxxx
15.2.4 Equations with |γ| + |δ| ̸= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2.5 Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′′′ = f (y)g(y ′ )h(y ′′ ) . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Equations of the Form yxxx
x
xx
15.3.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
15.3.3 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4 Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
15.4.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
15.4.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
15.4.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
15.5 Nonlinear Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . .
′′′ + G(x, y) = 0 . . . . . . . . . . . . . . . . .
15.5.1 Equations of the Form F (x, y)yxxx
′′′ + G(x, y, y ′ ) = 0 . . . . . . . . . . .
15.5.2 Equations of the Form F (x, y, yx′ )yxxx
x
′
′′′
′′ +H(x, y, y ′ )=
15.5.3 Equations of the Form F (x, y, yx )yxxx +G(x, y, yx′ )yxx
x
0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .∑
.........................
′′′ +
′
′′ α
15.5.4 Equations of the Form F (x, y, yx′ )yxxx
α Gα (x, y, yx )(yxx ) = 0
15.5.5 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
901
901
909
911
912
944
945
945
948
952
955
955
963
967
971
974
978
978
980
987
992
994
16 Fourth-Order Ordinary Differential Equations
16.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1.3 Equations Containing Exponential and Hyperbolic Functions . . . . . . .
16.1.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
16.1.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
16.1.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
16.2.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
16.2.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
16.2.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
16.2.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .
999
999
999
1000
1007
1011
1012
1015
1019
1019
1027
1029
1034
1035
1040
17 Higher-Order Ordinary Differential Equations
17.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.2 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.3 Equations Containing Exponential and Hyperbolic Functions . . . . . . .
17.1.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
17.1.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
17.1.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .
1051
1051
1051
1051
1058
1061
1061
1064
17.2 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.1 Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.2 Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . .
17.2.3 Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
17.2.4 Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . .
17.2.5 Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
17.2.6 Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .
1068
1068
1075
1077
1081
1083
1087
18 Some Systems of Ordinary Differential Equations
18.1 Linear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1.1 Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1.2 Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Linear Systems of Three and More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.1 Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.2 Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4 Nonlinear Systems of Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.1 Systems of Three Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.2 Dynamics of a Rigid Body with a Fixed Point . . . . . . . . . . . . . . . . . . . . .
1099
1099
1099
1102
1107
1108
1108
1110
1113
1113
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
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1.2 Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1.3 Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2.1 Exact Analytical Solutions in Terms of Predefined Functions . . . . . . . .
19.2.2 Exact Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . .
19.2.3 Different Types of Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2.4 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . .
19.2.5 Integral Transform Methods for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2.6 Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . .
19.3 Group Analysis of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3.1 Solution Strategies and Predefined Functions . . . . . . . . . . . . . . . . . . . . . .
19.3.2 Constructing Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3.3 Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3.4 Order Reduction of ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.4.1 Numerical Solutions in Terms of Predefined Functions . . . . . . . . . . . . .
19.4.2 Numerical Methods Embedded in Maple . . . . . . . . . . . . . . . . . . . . . . . . .
19.4.3 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . .
19.4.4 Initial Value Problems: Constructing Numerical Methods and
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.4.5 Boundary Value Problems: Examples of Numerical Solutions . . . . . . .
19.4.6 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . .
19.4.7 First-Order Systems of ODEs. Higher-Order ODEs. Numerical
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1121
1121
1121
1122
1124
1127
1127
1137
1142
1144
1147
1149
1153
1153
1155
1156
1158
1160
1160
1162
1168
1171
1176
1181
1183
20 Symbolic and Numerical Solutions of ODEs with Mathematica
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.1.1 Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.1.2 Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.1 Exact Analytical Solutions in Terms of Predefined Functions . . . . . . . .
20.2.2 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . .
20.2.3 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.4 Integral Transform Methods for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.5 Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . .
20.3 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3.1 Numerical Solutions in Terms of Predefined Functions . . . . . . . . . . . . .
20.3.2 Numerical Methods Embedded in Mathematica . . . . . . . . . . . . . . . . . . .
20.3.3 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . .
20.3.4 Initial Value Problems: Constructing Numerical Methods and
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3.5 Boundary Value Problems: Examples of Numerical Solutions . . . . . . .
20.3.6 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . .
20.3.7 First-Order Systems of ODEs. Higher-Order ODEs. Numerical
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3.8 Phase Plane Analysis for First-Order Autonomous Systems . . . . . . . . .
20.3.9 Numerical-Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1187
1187
1187
1189
1193
1193
1203
1207
1209
1212
1215
1215
1217
1221
21 Symbolic and Numerical Solutions of ODEs with MATLAB
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.1.2 Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.1.3 MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.2.1 Analytical Solutions in Terms of Predefined Functions . . . . . . . . . . . . .
21.2.2 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . .
21.2.3 Analytical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . .
21.3 Numerical Solutions of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.3.1 Numerical Solutions via Predefined Functions . . . . . . . . . . . . . . . . . . . . .
21.3.2 Initial Value Problems: Examples of Numerical Solutions . . . . . . . . . .
21.3.3 Boundary Value Problems: Examples of Numerical Solutions . . . . . . .
21.3.4 Eigenvalue Problems: Examples of Numerical Solutions . . . . . . . . . . . .
21.4 Numerical Solutions of Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.4.1 First-Order Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.4.2 First-Order Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1245
1245
1245
1246
1249
1253
1253
1259
1261
1263
1263
1268
1272
1276
1279
1279
1282
Part IV. Supplements
1285
S1 Elementary Functions and Their Properties
S1.1 Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.1.1 Properties of the Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.1.2 Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.1.3 Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
1287
1287
1287
1287
1288
1223
1229
1235
1236
1239
1241
S1.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.1 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.2 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.3 Relations between Trigonometric Functions of Single Argument . . . . .
S1.2.4 Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . .
S1.2.5 Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.6 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.8 Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . .
S1.2.9 Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . .
S1.2.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.2.13 Representation in the Form of Infinite Products . . . . . . . . . . . . . . . . . . .
S1.2.14 Euler and de Moivre Formulas. Relationship with Hyperbolic
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3.1 Definitions of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . .
S1.3.2 Simplest Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3.3 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3.4 Relations between Inverse Trigonometric Functions . . . . . . . . . . . . . . . .
S1.3.5 Addition and Subtraction of Inverse Trigonometric Functions . . . . . . .
S1.3.6 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3.7 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.3.8 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.1 Definitions of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.2 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.3 Relations between Hyperbolic Functions of Single Argument (x ≥ 0) .
S1.4.4 Addition and Subtraction of Hyperbolic Functions . . . . . . . . . . . . . . . . .
S1.4.5 Products of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.6 Powers of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.8 Hyperbolic Functions of Multiple Argument . . . . . . . . . . . . . . . . . . . . . .
S1.4.9 Hyperbolic Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.4.13 Representation in the Form of Infinite Products . . . . . . . . . . . . . . . . . . .
S1.4.14 Relationship with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . .
S1.5 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.5.1 Definitions of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . .
S1.5.2 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.5.3 Relations between Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . .
S1.5.4 Addition and Subtraction of Inverse Hyperbolic Functions . . . . . . . . . .
S1.5.5 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.5.6 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S1.5.7 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1289
1289
1290
1290
1290
1291
1291
1291
1292
1292
1292
1292
1293
1293
1293
1293
1293
1294
1294
1295
1295
1296
1296
1296
1296
1296
1297
1297
1297
1297
1298
1298
1298
1299
1299
1299
1299
1299
1300
1300
1300
1300
1300
1300
1301
1301
1301
S2 Indefinite and Definite Integrals
S2.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.1 Integrals Involving Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.2 Integrals Involving Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.3 Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.4 Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.5 Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . .
S2.1.6 Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . .
S2.1.7 Integrals Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . .
S2.2 Tables of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S2.2.1 Integrals Involving Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . .
S2.2.2 Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . .
S2.2.3 Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . .
S2.2.4 Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . .
S2.2.5 Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . .
S2.2.6 Integrals Involving Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3 Tables of Laplace and Inverse Laplace Transforms
S3.1 Tables of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.1 General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.2 Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.3 Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.4 Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.5 Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.6 Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .
S3.1.7 Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2 Tables of Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.1 General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.2 Expressions with Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.3 Expressions with Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.4 Expressions with Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.5 Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.6 Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.7 Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.8 Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .
S3.2.9 Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4 Special Functions and Their Properties
S4.1 Some Coefficients, Symbols, and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.1.1 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.1.2 Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.1.3 Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.1.4 Euler Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.2 Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . .
S4.2.1 Error Function and Complementary Error Function . . . . . . . . . . . . . . . .
S4.2.2 Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.2.3 Logarithmic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.3 Sine Integral and Cosine Integral. Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . .
S4.3.1 Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.3.2 Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.3.3 Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1303
1303
1303
1307
1310
1311
1314
1315
1319
1320
1320
1323
1324
1325
1325
1328
1331
1331
1331
1333
1333
1334
1335
1335
1337
1338
1338
1340
1344
1345
1346
1347
1348
1349
1349
1351
1351
1351
1352
1352
1353
1354
1354
1354
1355
1356
1356
1357
1357
S4.4 Gamma Function, Psi Function, and Beta Function . . . . . . . . . . . . . . . . . . . . . . .
S4.4.1 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.4.2 Psi Function (Digamma Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.4.3 Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.5 Incomplete Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.5.1 Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.5.2 Incomplete Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.6 Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.6.1 Definitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.6.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . .
S4.6.3 Zeros and Orthogonality Properties of Bessel Functions . . . . . . . . . . . .
S4.6.4 Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . .
S4.7 Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.7.1 Definitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.7.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . .
S4.8 Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.8.1 Definition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.8.2 Power Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . .
S4.9 Degenerate Hypergeometric Functions (Kummer Functions) . . . . . . . . . . . . . . .
S4.9.1 Definitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.9.2 Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . .
S4.9.3 Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.10 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.10.1 Various Representations of the Hypergeometric Function . . . . . . . .
S4.10.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.11 Legendre Polynomials, Legendre Functions, and Associated Legendre
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.11.1 Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . .
S4.11.2 Associated Legendre Functions with Integer Indices and Real
Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.11.3 Associated Legendre Functions. General Case . . . . . . . . . . . . . . . . .
S4.12 Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.12.1 Definitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.12.2 Integral Representations, Asymptotic Expansions, and Linear
Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.13 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.13.1 Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.13.2 Incomplete Elliptic Integrals (Elliptic Integrals) . . . . . . . . . . . . . . . .
S4.14 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.14.1 Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.14.2 Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.15 Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.15.1 Series Representation of the Jacobi Theta Functions. Simplest
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.15.2 Various Relations and Formulas. Connection with Jacobi Elliptic
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.16 Mathieu Functions and Modified Mathieu Functions . . . . . . . . . . . . . . . . . . . .
S4.16.1 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.16.2 Modified Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1358
1358
1359
1360
1360
1360
1361
1362
1362
1364
1366
1367
1367
1367
1369
1370
1370
1370
1371
1371
1374
1375
1375
1375
1377
1377
1377
1380
1380
1383
1383
1384
1385
1385
1386
1388
1388
1392
1394
1394
1395
1396
1396
1398
S4.17 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.17.1 Laguerre Polynomials and Generalized Laguerre Polynomials . . . .
S4.17.2 Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . .
S4.17.3 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.17.4 Jacobi Polynomials and Gegenbauer Polynomials . . . . . . . . . . . . . . .
S4.18 Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.18.1 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S4.18.2 Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Index
1398
1398
1400
1402
1404
1405
1405
1406
1409
1427
REFERENCES
Abel, M. L. and Braselton, J. P., Maple by Example, 3rd ed., AP Professional, Boston, MA, 2005.
Abramowitz, M. and Stegun, I. A. (Editors), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics,
Washington, 1964.
Acosta, G., Durán, G., and Rossi, J. D., An adaptive time step procedure for a parabolic problem
with blow-up, Computing, Vol. 68, pp. 343–373, 2002.
Akô, K., Subfunctions for ordinary differential equations, II, Funkcialaj Ekvacioj (International
Series), Vol. 10, No. 2, pp. 145–162, 1967.
Akô, K., Subfunctions for ordinary differential equations, III, Funkcialaj Ekvacioj (International
Series), Vol. 11, No. 2, pp. 111–129, 1968.
Akritas, A. G., Elements of Computer Algebra with Applications, Wiley, New York, 1989.
Akulenko, L. D. and Nesterov, S. V., Accelerated convergence method in the Sturm–Liouville
problem, Russ. J. Math. Phys., Vol. 3, No. 4, pp. 517–521, 1996.
Akulenko, L. D. and Nesterov, S. V., Determination of the frequencies and forms of oscillations of
non-uniform distributed systems with boundary conditions of the third kind, Appl. Math. Mech.
(PMM), Vol. 61, No. 4, pp. 531–538, 1997.
Akulenko, L. D. and Nesterov, S. V., High Precision Methods in Eigenvalue Problems and Their
Applications, Chapman & Hall/CRC Press, Boca Raton, 2005.
Alexandrov, A. Yu., Platonov, A. V., Starkov, V. N., Stepenko, N. A., Mathematical Modeling
and Stability Analysis of Biological Communities, Lan’, St. Petersburg, 2016.
Alexeeva, T. A., Zaitsev, V. F., and Shvets, T. B., On discrete symmetries of the Abel equation of
the 2nd kind [in Russian]. In: Applied Mechanics and Mathematics, MIPT, Moscow, pp. 4–11,
1992.
Alshina, E. A., Kalitkin, N. N., and Koryakin, P. V., Diagnostics of singularities of exact solutions
in computations with error control [in Russian], Zh. Vychisl. Mat. Mat. Fiz., Vol. 45, No. 10,
pp. 1837–1847, 2005; http://eqworld.ipmnet.ru/ru/solutions/interesting/alshina2005.pdf.
Anderson, R. L. and Ibragimov, N. H., Lie–Bäcklund Transformations in Applications, SIAM
Studies in Applied Mathematics, Philadelphia, 1979.
Andronov, A. A., Vitt, A. A., and Khaikin, S. E., Theory of Oscillators, Dover Publ., New York,
2011.
Andronov, A. A., Leontovich, E. A., and Gordon, I. I., Theory of Bifurcations of Dynamic Systems
on a Plane, Israel Program for Scientific Translations (IPST), Jerusalem, 1971.
Antimirov, M. Ya., Applied Integral Transforms, American Mathematical Society, Providence,
Rhode Island, 1993.
Arnold, V. I., Additional Chapters of Ordinary Differential Equation Theory [in Russian], Nauka,
Moscow, 1978.
Arnold, V. I. and Afraimovich, V. S., Bifurcation Theory and Catastrophe Theory, SpringerVerlag, New York, 1999.
Arnold, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial
Mechanics, Dynamical System III, Springer, Berlin, 1993.
Ascher, U., Mattheij, R. and Russell, R., Numerical Solution of Boundary Value Problems for
Ordinary Differential Equations, Ser. SIAM Classics in Applied Mathematics, Vol. 13, 1995.
Ascher, U. and Petzold, L., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1998.
1409
Ascher, U. and Petzold, L.,. Projected implicit Runge–Kutta methods for differential algebraic
equations, SIAM J. Numer. Anal., Vol. 28, pp. 1097–1120, 1991.
Bader, G. and Deuflhard, P., A semi-implicit mid-point rule for stiff systems of ordinary
differential equations, Numer. Math, Vol. 41, pp. 373–398, 1983.
Bahder, T. B., Mathematica for Scientists and Engineers, Addison-Wesley, Redwood City, CA,
1994.
Bailey, P. B., Shampine, L. F., and Waltman, P. E., Nonlinear Two Point Boundary Value
Problems, Academic Press, New York, 1968.
Baker, G. A. (Jr.) and Graves–Morris, P., Padé Approximants, Addison–Wesley, London, 1981.
Bakhvalov, N. S., Numerical Methods: Analysis, Algebra, Ordinary Differential Equations, Mir
Publishers, Moscow, 1977.
Barton, D., Willer, I. M. and Zahar, R. V. M., Taylor series method for ordinary differential
equations. In: Mathematical Software (J. R. Rice, editor), Academic Press, New York, 1972.
Barton, D., Willer, I. M., and Zahar, R. V. M., The automatic solution of systems of ordinary
differential equations by the method of Taylor series, Comput. J., Vol. 14, pp. 243–248, 1971.
Bateman, H. and Erdélyi, A., Higher Transcendental Functions, Vols. 1 and 2, McGraw-Hill, New
York, 1953.
Bateman, H. and Erdélyi, A., Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York,
1955.
Bateman, H. and Erdélyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill, New
York, 1954.
Bazykin, A. D., Mathematical Biophysics of Interacting Populations [in Russian], Nauka, Moscow,
1985.
Bebernes, J. and Gaines, R., Dependence on boundary data and a generalized boundary value
problem, J. Differential Equations, Vol. 4, pp. 359–368, 1968.
Bekir, A., New solitons and periodic wave solutions for some nonlinear physical models by using
the sine-cosine method, Physica Scripta, Vol. 77, No. 4, 2008.
Bekir, A., Application of the G′ /G-expansion method for nonlinear evolution equations, Phys. Lett.
A, Vol. 372, pp. 3400–3406, 2008.
Bellman, R. and Roth, R., The Laplace Transform, World Scientific Publishing Co., Singapore,
1984.
Berezin, I. S. and Zhidkov, N. P., Computational Methods, Vol. II [in Russian], Fizmatgiz,
Moscow, 1960.
Berkovich, L. M., Factorization and Transformations of Ordinary Differential Equations [in Russian], Saratov University Publ., Saratov, 1989.
Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, John Wiley & Sons, New York,
1978.
Blaquiere, A., Nonlinear System Analysis, Academic Press, New York, 1966.
Bluman, G. W. and Anco, S. C., Symmetry and Integration Methods for Differential Equations,
Springer, New York, 2002.
Bluman, G. W. and Cole, J. D., Similarity Methods for Differential Equations, Springer, New York,
1974.
Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer, New York, 1989.
Bogacki, P. and Shampine, L. F., An Efficient Runge–Kutta (4, 5) Pair, Report 89–20, Math. Dept.
Southern Methodist University, Dallas, Texas, 1989.
Bogolyubov, N. N. and Mitropol’skii, Yu. A., Asymptotic Methods in the Theory of Nonlinear
Oscillations [in Russian], Nauka, Moscow, 1974.
1410
Boltyanskii, V. G. and Vilenkin, N. Ya., Symmetry in Algebra, 2nd ed. [in Russian], Nauka,
Moscow, 2002.
Boor, C. and Swartz, B., SIAM J. Numerical Analysis, Vol. 10, No. 4, pp. 582–606, 1993.
Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics [in Russian], Regular and Chaotic
Dynamics, Izhevsk, 2001.
Boyce, W. E. and DiPrima, R. C., Elementary Differential Equations and Boundary Value
Problems, 8th Edition, John Wiley & Sons, New York, 2004.
Bratus, A. S., Novozhilov, A. S., and Platonov, A. P., Dynamic Models and Models in Biology [in
Russian], Fizmatlit, Moscow, 2009.
Braun, M., Differential Equations and Their Applications, 4th Edition, Springer, New York, 1993.
Brenan, K. E., Campbell, S. L., and Petzold, L. R., Numerical Solution of Initial-Value Problems
in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996.
Brent, R. P., Algorithms for Minimization without Derivatives, Dover, 2002 (Original edition 1973).
Bulirsch, R. and Stoer, J., Fehlerabschätzungen und extrapolation mit rationalen funktionen bei
verfahren vom Richardson–typus, Numer. Math., Vol. 6, pp. 413–427, 1964.
Butcher, J. C., Modified multistep method for numerical integration of ordinary differential
equations, J. Assoc. Comput. Mach., Vol. 12, No. 1, pp. 124–135, 1965.
Butcher, J. C., Order, stepsize and stiffness switching, Computing, Vol. 44, No. 3, pp. 209–220,
1990.
Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and
General Linear Methods, Wiley-Interscience, New York, 1987.
Candy, J. and Rozmus, R., A symplectic integration algorithm for separable Hamiltonian functions, J. Comput. Phys., Vol. 92, pp. 230–256, 1991.
Cash, J. R., The integration of stiff IVP in ODE using modified extended BDF, Computers and
Mathematics with Applications, Vol. 9, pp. 645–657, 1983.
Cash, J. R. and Considine, S., An MEBDF code for stiff initial value problems, ACM Trans. Math.
Software (TOMS), Vo. 18, No. 2, pp. 142–155, 1992.
Cash, J. R. and Karp, A. H., A variable order Runge–Kutta method for initial value problems with
rapidly varying right-hand sides, ACM Transactions on Mathematical Software, Vol. 16, No. 3,
pp. 201–222, 1990.
Chapra, S. C. and Canale, R. P., Numerical Methods for Engineers, 6th Edition, McGraw-Hill,
Boston, 2010.
Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., and Watt, S. M., A Tutorial
Introduction to Maple V, Springer, Wien, New York, 1992.
Cheb-Terrab, E. S., Duarte, L. G. S., and da Mota, L. A. C. P., Computer algebra solving of first
order ODEs using symmetry methods, Computer Physics Communications, Vol. 101, pp.254–
268, 1997.
Cheb-Terrab, E. S. and von Bulow, K., A computational approach for the analytical solving of
partial differential equations, Computer Physics Communications, Vol. 90, pp. 102–116, 1995.
Cheb-Terrab, E. S. and Kolokolnikov, T., First-order ordinary differential equations, symmetries
and linear transformations, European Journal of Applied Mathematics, Vol. 14, pp. 231–246,
2003.
Cheb-Terrab, E. S. and Roche, A. D., Symmetries and first order ODE patterns, Computer Physics
Communications, Vol. 113, pp. 239–260, 1998.
Cheb-Terrab, E. S., Duarte, L. G. S., and da Mota, L. A. C. P., Computer algebra solving of
second order ODEs using symmetry methods, Computer Physics Communications, Vol. 108,
pp. 90–114, 1998.
1411
Chicone, C., Ordinary Differential Equations with Applications, Springer, Berlin, 1999.
Chowdhury, A. R., Painlevé Analysis and Its Applications, Chapman & Hall/CRC Press, Boca
Raton, 2000.
Clarkson, P. A., The third Painlevé equation and associated special polynomials, J. Phys. A, Vol.
36, pp. 9507–9532, 2003.
Clarkson, P. A., Painlevé Equations — Nonlinear Special Functions: Computation and Application.
In: Orthogonal Polynomials and Special Functions (Eds. Marcell, F. and van Assche, W.), Vol.
1883 of Lecture Notes in Math., pp. 331–411. Springer, Berlin , 2006.
Cohen, S. D. and Hindmarsh, A. C., CVODE, a Stiff/Nonstiff ODE Solver in C, Computers in
Physics, Vol. 10, No. 2, pp. 138–143, 1996.
Cole, G. D., Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company,
Waltham, MA, 1968.
Collatz, L., Eigenwertaufgaben mit Technischen Anwendungen, Akademische Verlagsgesellschaft
Geest & Portig, Leipzig, 1963.
Conte, S. D. and de Boor, C., Elementary Numerical Analysis. An Algorithmic Approach, McGrawHill, 1980.
Conte, R., The Painlevé approach to nonlinear ordinary differential equations. In: The Painlevé
Property (ed. by R. Conte), pp. 77–180, CRM Series in Mathematical Physics, Springer, New
York, 1999.
Corless, R. M., Essential Maple, Springer, Berlin, 1995.
Crawford, J. D., Introduction to bifurcation theory, Rev. Mod. Phys., Vol. 63, No. 4, 1991.
Davenport, J. H., Siret, Y., and Tournier, E., Computer Algebra Systems and Algorithms for
Algebraic Computation, Academic Press, London, 1993.
Dekker, T. J., Finding a zero by means of successive linear interpolation. In: Constructive Aspects
of the Fundamental Theorem of Algebra (Dejon, B. and Henrici, P., eds.), Wiley-Interscience,
London, 1969.
Del Buono, N. and Lopez, L., Runge–Kutta type methods based on geodesics for systems of ODEs
on the Stiefel manifold, BIT, Vol. 41, No. 5, pp. 912–923, 2001.
Deuflhard, P., Recent progress in extrapolation methods for ordinary differential equations, SIAM
Rev., Vol. 27, pp. 505–535, 1985.
Deuflhard, P., Hairer, E. and Zugck, J., One-step and extrapolation methods for differentialalgebraic systems, Numer. Math., Vol. 51, pp. 501–516, 1987.
Deuflhard, P., Fiedler, B., and Kunkel, P., Efficient numerical path following beyond critical
points, SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics,
Vol. 24, pp. 912–927, 1987.
Dieci, L., Russel, R. D., and van Vleck, E. S., Unitary integrators and applications to continuous
orthonormalization techniques, SIAM J. Num. Anal., Vol 31, pp. 261–281, 1994.
Dieci, L. and van Vleck, E. S., Computation of orthonormal factors for fundamental solution
matrices, Numer. Math., Vol. 83, pp. 599–620, 1999.
Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon
Press, New York, 1965.
Dlamini, P.G. and Khumalo, M., On the computation of blow-up solutions for semilinear ODEs
and parabolic PDEs, Math. Problems in Eng., Vol. 2012, Article ID 162034, 15 p., 2012.
Dobrokhotov, S. Yu., Integration by quadratures of 2n-dimensional linear Hamiltonian systems
with n known skew-orthogonal solutions, Russ. Math. Surveys, Vol. 53, No. 2, pp. 380–381,
1998.
Doetsch, G., Handbuch der Laplace-Transformation. Theorie der Laplace-Transformation, Birkhäuser, Basel–Stuttgart, 1950.
1412
Doetsch, G., Handbuch der Laplace-Transformation. Anwendungen der Laplace-Transformation,
Birkhäuser, Basel–Stuttgart, 1956.
Doetsch, G., Introduction to the Theory and Application of the Laplace Transformation, Springer,
Berlin, 1974.
Dormand, J. R., Numerical Methods for Differential Equations: A Computational Approach, CRC
Press, Boca Raton, 1996.
Ebaid, A., Exact solitary wave solutions for some nonlinear evolution equations via Exp-function
method, Phys. Letters A, Vol. 365, No. 3, pp. 213–219, 2007.
Elkin, V. I., Reduction of underdetermined systems of ordinary differential equations: I, Differential
Equations, Vol. 45, No. 12, pp. 1721–1731, 2009.
Elkin, V. I., Classification of certain types of underdetermined systems of ordinary differential
equations, Doklady Mathematics, Vol. 81, No. 3, pp. 362–363, 2010.
El’sgol’ts, L. E., Differential Equations, Gordon & Breach Inc., New York, 1961.
Enright, W. H., The Relative Efficiency of Alternative Defect Control Schemes for High Order
Continuous Runge–Kutta Formulas, Technical Report 252/91, Dept. of Computer Science,
University of Toronto, 1991.
Enright, W. H., Jackson, K. R., Nørsett, S. P. and Thomsen, P. G., Interpolants for Runge–Kutta
formulas, ACM TOMS, Vol. 12, pp. 193–218, 1986.
Enright, W. H., Jackson, K. R., Nørsett, S. P. and Thomsen, P. G., Effective solution of
discontinuous IVPS using a RKF pair with interpolants, Proceed. ODE Conference held at
Sandia National Lab., Albuquerque, New Mexico, 1986.
Enright, W. H., A new error-control for initial value solvers, Appl. Math. Comput., Vol. 31, pp. 588–
599, 1989.
Erbe, L. H., Hu, S., and Wang, H., Multiple positive solutions of some boundary value problems,
J. Math. Anal. Appl., Vol. 184, pp. 640–648, 1994.
Erbe, L. H. and Wang, H.,, On the existence of positive solutions of ordinary differential equations,
Proc. Amer. Math. Soc., Vol. 120, pp. 743–748, 1994.
Evans, D. J and Raslan, K. R., The Adomian decomposition method for solving delay differential
equations, Int. J. Comput. Math., Vol. 82, pp. 49–54, 2005.
Faddeev, S. I. and Kogan, V. V., Nonlinear Boundary Value Problems for Systems of Ordinary Differential Equations on Finite Interval [in Russian], Novosibirsk State University, Novosibirsk,
2008.
Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Letters
A, Vol. 277, No. 4–5, pp. 212–218, 2000.
Fedoryuk, M. V., Asymptotic Analysis. Linear Ordinary Differential Equations, Springer, Berlin,
1993.
Fehlberg, E., Klassische Runge–Kutta–Formeln vierter und niedrigerer ordnung mit schrittweitenlontrolle und ihre anwendung auf waermeleitungsprobleme, Computing, Vol. 6, pp 61–71, 1970.
Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic Press,
New York, 1972.
Fokas, A. S. and Ablowitz M. J., On unified approach to transformation and elementary solutions
of Painlevé equations, J. Math. Phys., Vol. 23, No. 11, pp. 2033–2042, 1982.
Forsythe, G. E., Malcolm, M. A., and Moler, C. B., Computer Methods for Mathematical
Computations, Prentice Hall, New Jersey, 1977.
Fox, L. and Mayers, D. F., Numerical Solution of Ordinary Differential Equations for Scientists
and Engineers, Chapman & Hall, 1987.
Frank-Kamenetskii, D. A., Diffusion and Heat Transfer in Chemical Kinetics, 2nd Edition [in
Russian], Nauka, Moscow, 1987 (English edition: Plenum Press, 1969).
1413
Gaisaryan, S. S., Differential equations, ordinary, approximate methods of solution. In: Encyclopedia of Mathematics, Kluwer, 2002. URL: http://www.encyclopediaofmath.org/index.php?title=
Differential equations, ordinary, approximate methods of solution of&oldid=11532
Galaktionov, V. A. and Svirshchevskii, S. R., Exact Solutions and Invariant Subspaces of
Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC
Press, Boca Raton, 2006.
Gambier, B., Sur les équations differentielles du second ordre et du premier degré dont l‘integrale
générale est à points critiques fixes, Acta Math., Vol. 33, pp. 1–55, 1910.
Gantmakher, F. R., Lectures on Analytical Mechanics [in Russian], Fizmatlit, Moscow, 2002.
Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall,
Upper Saddle River, NJ, 1971.
Geddes, K. O., Czapor, S. R., and Labahn, G., Algorithms for Computer Algebra, Kluwer
Academic Publishers, Boston, 1992.
Getz, C. and Helmstedt, J., Graphics with Mathematica: Fractals, Julia Sets, Patterns and Natural
Forms, Elsevier Science & Technology Book, Amsterdam, Boston, 2004.
Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Springer, New York, 1972.
Godunov, S. K. and Ryaben’kii, V. S., Differences Schemes [in Russian], Nauka, Moscow, 1973.
Golub, G. H. and van Loan, C. F., Matrix Computations, 3rd ed., Johns Hopkins University Press,
1996.
Golubev, V. V., Lectures on Analytic Theory of Differential Equations [in Russian], GITTL,
Moscow, 1950.
Goriely, A. and Hyde, C., Necessary and sufficient conditions for finite time singularities in
ordinary differential equations, J. Differential Equations, Vol. 161, pp. 422–448, 2000.
Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series, and Products, Academic Press,
New York, 1980.
Gragg, W. B., On extrapolation algorithms for ordinary initial value problems, SIAM J. Num. Anal.,
Vol. 2, pp. 384–403, 1965.
Gray, T. and Glynn, J., Exploring Mathematics with Mathematica: Dialogs Concerning Computers and Mathematics, Addison-Wesley, Reading, MA, 1991.
Gray, J. W., Mastering Mathematica: Programming Methods and Applications, Academic Press,
San Diego, 1994.
Green, E., Evans, B. and Johnson, J., Exploring Calculus with Mathematica, Wiley, New York,
1994.
Grimshaw, R., Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, 1991.
Gromak, V. I., Painlevé Differential Equations in the Complex Plane, Walter de Gruyter, Berlin,
2002.
Gromak, V. I. and Lukashevich, N. A., Analytical Properties of Solutions of Painlevé Equations
[in Russian], Universitetskoe, Minsk, 1990.
Guckenheimer J. and Holms P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of
Vector Fields, Springer-Verlag, New York, 1983.
Gustafsson, K., Control theoretic techniques for stepsize selection in explicit Runge–Kutta methods, ACM Trans. Math. Soft., No. 17, pp. 533–554, 1991.
Hairer, E. and Lubich, C., On extrapolation methods for stiff and differential-algebraic equations,
Teubner Texte zur Mathematik, Vol. 104, pp. 64–73, 1988.
Hairer, E., Symmetric projection methods for differential equations on manifolds, BIT, Vol. 40,
No. 4, pp. 726–734, 2000.
1414
Hairer, E., Lubich, C., and Roche, M., The Numerical Solution of Differential-Algebraic Systems
by Runge-Kutta Methods, Springer, Berlin, 1989.
Hairer, E., Lubich, C., and Wanner, G., Geometric Numerical Integration: Structure-Preserving
Algorithms for Ordinary Differential Equations, Springer, Berlin, 2002.
Hairer, E., Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff
Problems, 2nd ed., Springer, Berlin, 1993.
Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, 2nd ed., Springer, New York, 1996.
Hartman, P., Ordinary Differential Equations, John Wiley & Sons, New York, 1964.
He, J.-H. and Wu, X.-H., Exp-function method for nonlinear wave equations, Chaos, Solitons &
Fractals, Vol. 30, No. 3, pp. 700–708, 2006.
He, J.-H. and Abdou, M. A., New periodic solutions for nonlinear evolution equations using Expfunction method, Chaos, Solitons & Fractals, Vol. 34, No. 5, pp. 1421–1429, 2007.
Heck, A., Introduction to Maple, 3rd ed., Springer, New York, 2003.
Higham, N. J., Functions of Matrices. Theory and Computation, SIAM, Philadelphia, 2008.
Hill, J. M., Solution of Differential Equations by Means of One-Parameter Groups, Pitman,
Marshfield, MA, 1982.
Hindmarsh, A. C., Odepack, a systemized collection of ODE solvers. In: Scientific Computing
(Stepleman, R. S. et al., eds.), pp. 55–64, North-Holland, Amsterdam, 1983.
Hirota, C. and Ozawa, K., Numerical method of estimating the blow-up time and rate of the
solution of ordinary differential equations: An application to the blow-up problems of partial
differential equations, J. Comput. & Applied Math., Vol. 193, No. 2, pp. 614-637, 2006.
Hosea, M. E. and Shampine, L. F., Analysis and implementation of TR-BDF2, Appl. Numer.
Math., Vol. 20, pp. 21–37, 1996.
Hubbard J. H. and West, B. H., Differential Equations: A Dynamical Systems Approach. Part I.
One Dimensional Equations, Springer, New York, 1990.
Hull, T. E., Enright, W. H., Fellen, B. M., and Sedgwick, A. E., Comparing numerical methods
for ordinary differential equations, SIAM J. Numer. Anal., Vol. 9, pp. 603–637, 1972.
Hydon, P. E., Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge Univ.
Press, Cambridge, 2000.
Ibragimov, N. H., A Practical Course in Differential Equations and Mathematical Modelling,
Higher Education Press – World Scientific Publ., Beijing – Singapore, 2010.
Ibragimov, N. H. (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1,
CRC Press, Boca Raton, 1994.
Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations, John
Wiley, Chichester, 1999.
Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, (English translation
published by D. Reidel), Dordrecht, 1985.
Ince, E. L., Ordinary Differential Equations, Dover Publications, New York, 1956.
Iooss G. and Joseph D. D., Elementary Stability and Bifurcation Theory, Springer-Verlag, New
York, 1997.
Its, A. R. and Novokshenov, V. Yu., The Isomonodromic Deformation Method in the Theory of
Painlevé Equations, Springer, Berlin, 1986.
Izhikevich, E. M., Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,
MIT Press, 2007.
Jackson, L. K. and Palamides, P. K.,, An existence theorem for a nonlinear two-point boundary
value problem, J. Differential Equations, Vol. 53, pp. 48–66, 1984.
1415
Jacobi, C. G. J., Vorlesungen über Dynamik, G. Reimer, Berlin, 1884.
Kahaner, D., Moler, C., and Nash, S., Numerical Methods and Software, Prentice-Hall, New
Jersey, 1989.
Kalitkin, N. N., Numerical Methods [in Russian], Nauka, Moscow, 1978.
Kalitkin, N. N., Alshin, A. B., Alshina, E. A., and Rogov, B. V., Computation on Quasiuniform
Meshes [in Russian], Fizmatlit, Moscow, 2005.
Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen, I, Gewöhnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977.
Kantorovich, L. V. and Akilov, G. P., Functional Analysis in Normed Spaces [in Russian],
Fizmatgiz, Moscow, 1959.
Kantorovich, L. V. and Krylov, V. I., Approximate Methods of Higher Analysis [in Russian],
Fizmatgiz, Moscow, 1962.
Karakostas, G. L., Nonexistence of solutions to some boundary-value problems for second-order
ordinary differential equations, Electron. J. Differential Equations, No. 20, pp. 1–10, 2012.
Keller, H. B., Numerical Solutions of Two Point Boundary Value Problems, SIAM, Philadelphia,
1976.
Keller, J. B., The shape of the strongest column, Archive for Rational Mechanics and Analysis,
Vol. 5, 275–285, 1960.
Keller J. and Antman S. (Eds.), Bifurcation Theory and Nonlinear Eigenvalue Problems, W. A.
Benjamin Publ., New York, 1969.
Kevorkian, J. and Cole, J. D., Perturbation Methods in Applied Mathematics, Springer, New York,
1981.
Kevorkian, J. and Cole, J. D., Multiple Scale and Singular Perturbation Methods, Springer, New
York, 1996.
Kierzenka, J. and Shampine, L. F., A BVP solver based on residual control and the MATLAB
PSE, ACM Trans. Math. Software, Vol. 27, pp. 299–316, 2001.
Klein, F. and Sommerfeld, A., Über die Theorie des Kreisels, Johnson Reprint corp., New York,
1965.
Klimov, D. M. and Zhuravlev, V. Ph., Group-Theoretic Methods in Mechanics and Applied
Mathematics, Taylor & Francis, London, 2002.
Korman, P. and Li, Y., Computing the location and the direction of bifurcation for sign changing
solutions, Dif. Equations & Applications, Vol. 2, No. 1, pp. 1–13, 2010.
Korman, P. and Li, Y., On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc.,
Vol. 127, No. 4, pp. 1011–1020, 1999.
Korman, P., Li, Y., and Ouyang, T., Computing the location and the direction of bifurcation, Math.
Research Letters, Vol. 12, pp. 933–944, 2005.
Korman, P., Global solution branches and exact multiplicity of solutions for two point boundary
value problems. In: Handbook of Differential Equations, Ordinary Differential Equations, Vol 3
(Canada A., Drabek P., and Fonda, A., eds.), Elsevier Science, North Holland, pp. 547–606,
2006.
Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd Edition,
Dover Publications, New York, 2000.
Kostyuchenko, A. G. and Sargsyan, I. S., Distribution of Eigenvalues (Self-Adjoint Ordinary
Differential Operators) [in Russian], Nauka, Moscow, 1979.
Kowalewsky, S., Sur le proble’me de la rotation d’un corps solide autor d’un point fixe, Acta. Math.,
Vol. 12, No. 2, pp. 177–232, 1889.
1416
Kowalewsky, S., Me’moires sur un cas particulies du proble’me de la rotation d’un point fixe, cu’
l’integration s’effectue a’ l’aide de fonctions ultraelliptiques du tems, Me’moires pre’sente’s par
divers savants a’ l’Acade’mie des seiences de l’Institut national de France, Paris, Vol. 31, pp.
1–62, 1890.
Koyalovich, B. M., Studies on the differential equation y dy−y dx = R dx [in Russian], Akademiya
Nauk, St. Petersburg, 1894.
Krasnosel’skii, M. A., Vainikko, G. M., Zabreiko, P. P., et al., Approximate Solution of Operator
Equations [in Russian], Nauka, Moscow, 1969.
Kreyszig, E., and Normington, E. J., Maple Computer Manual for Advanced Engineering
Mathematics, Wiley, New York, 1994.
Kudryashov, N. A., Symmetry of algebraic and differential equations, Soros Educational Journal
[in Russian], No. 9, pp. 104–110, 1998.
Kudryashov, N. A., Nonlinear differential equations with exact solutions expressed via the
Weierstrass function, Zeitschrift fur Naturforschung, Vol. 59, pp. 443–454, 2004.
Kudryashov, N. A., Analytical Theory of Nonlinear Differential Equations [in Russian], Institut
kompjuternyh issledovanii, Moscow–Izhevsk, 2004.
Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential
equations, Chaos, Solitons and Fractals, Vol. 24, No. 5, pp. 1217–1231, 2005.
Kudryashov, N. A., A note on the G′ /G-expansion method, Applied Math. & Computation,
Vol. 217, No. 4, pp. 1755–1758, 2010.
Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations,
Commun. Nonlinear Sci. Numer. Simulat., Vol. 17, pp. 2248–2253, 2012.
Kudryashov, N. A., Polynomials in logistic function and solitary waves of nonlinear differential
equations, Applied Math. & Computation, Vol. 219, pp. 9245–9253, 2013.
Kudryashov, N. A., Logistic function as solution of many nonlinear differential equations, Applied
Math. & Modelling, Vol. 39, No. 18, pp. 5733–5742, 2015.
Kudryashov, N. A. and Loguinova, N. V., Extended simplest equation method for nonlinear
differential equations, Applied Math. & Computation, Vol. 205, pp. 396–402, 2008.
Kudryashov, N. A. and Sinelshchikov, D. I., Nonlinear differential equations of the second, third
and fourth order with exact solutions, Applied Math. & Computation, Vol. 218, pp. 10454–
10467, 2012.
Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, 3rd Edition, Springer, New York, 2004.
Laetsch, T., The number of solutions of a nonlinear two point boundary value problem, Indiana
Univ. Math. J., Vol. 20, pp. 1–13, 1970.
Lagerstrom, P. A., Matched Asymptotic Expansions. Ideas and Techniques, Springer, New York,
1988.
Lagrange, J. L., Me’canique analyticque. Oeuvres de Lagrange, Vol. 12, Gauthier–Villars, Paris,
1889.
Lambert, J. D., Computational Methods in Ordinary Differential Equations, Wiley, New York,
1973.
Lambert, J. D., Numerical Methods for Ordinary Differential Systems, Wiley, New York, 1991.
Lapidus, L., Aiken, R. C., and Liu, Y. A., The occurrence and numerical solution of physical and
chemical systems having widely varying time constants. In: Stiff Differential Systems (R. A.
Willoughby, editor), pp. 187–200, Plenum, New York, 1973.
Lee, H. J. and Schiesser, W. E., Ordinary and Partial Differential Equation Routines in C, C++,
Fortran, Java, Maple, and MATLAB, Chapman & Hall/CRC Press, Boca Raton, 2004.
LePage, W. R., Complex Variables and the Laplace Transform for Engineers, Dover Publications,
New York, 1980.
1417
Levitan, B. M. and Sargsjan, I. S., Sturm–Liouville and Dirac Operators, Kluwer Academic,
Dordrecht, 1990.
Lian, W.-C., Wong, F.-H., and Yen, C.-C., On the existence of positive solutions of nonlinear
second order differential equations, Proc. American Math. Society, Vol. 124, No. 4, pp. 1117–
1126, 1996.
Lin, C. C. and Segel, L. A., Mathematics Applied to Deterministic Problems in the Natural
Sciences, SIAM, Philadelphia, PA, 1998.
Linchuk, L. V., On group analysis of functional differential equations [in Russian]. In: Proc. of
the Int. Conf MOGRAN-2000 “Modern Group Analysis for the New Millennium,” pp. 111–115,
USATU Publishers, Ufa, 2001.
Linchuk, L. V. and Zaitsev, V. F., Searching for first integrals and alternative symmetries
[in Russian]. In: Some Topical Problems of Modern Mathematics and Mathematical Education,
Proc. LXVIII International Conference “Herzen Readings – 2015” (13–17 April 2015, St.
Petersburg, Russia), A. I. Herzen Russian State Pedagogical University, 2015, pp. 50–53.
Lubich, C., Linearly implicit extrapolation methods for differential-algebraic systems, Numer.
Math., Vol. 55, pp. 197–211, 1989.
MacCallum, M. A. H., Using computer algebra to solve ordinary differential equations. In: Studies
for Computer Algebra in Industry, (Cohen, A.M., van Gastel L. J. and Verduyn Lunel, S. M.,
eds.) Vol. 2, John Wiley and Sons, Chichester, 1995.
MacDonald, N., Biological Delay Systems: Linear Stability Theory, Cambridge University Press,
Cambridge, 1989.
Maeder, R. E., Programming in Mathematica, 3rd ed., Addison-Wesley, Reading, MA, 1996.
Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am. J. Phys., Vol. 60, No. 7,
pp. 650–654, 1992.
Malfliet, W. and Hereman, W., The tanh method: exact solutions of nonlinear evolution and wave
equations, Phys. Scripta, Vol. 54, pp. 563–568, 1996.
Maplesoft, Differential Equations in Maple 15, Maplesoft, Waterloo, 2012.
Marchenko, V. A., Sturm–Liouville Operators and Applications, Birkhäuser Verlag, Basel–Boston,
1986.
Marchuk, G., Some applications of splitting-up methods to the solution of mathematical physics
problems, Aplikace Matematiky, Vol. 13, pp. 103–132, 1968.
Markeev, A. P., Theoretical Mechanics [in Russian], Regular and Chaotic Dynamics, Moscow–
Izhevsk, 2001.
Marsden J. E. and McCracken M., Hopf Bifurcation and Its Applications, Springer-Verlag, New
York, 1976.
Matveev, N. M., Methods of Integration of Ordinary Differential Equations [in Russian], Vysshaya
Shkola, Moscow, 1967.
McGarvey, J. F., Approximating the general solution of a differential equation, SIAM Review,
Vol. 24, No. 3, pp. 333–337, 1982.
McLachlan, N. W., Theory and Application of Mathieu Functions, Clarendon Press, Oxford, 1947.
McLachlan, R. I. and Atela, P., The accuracy of symplectic integrators, Nonlinearity, Vol. 5,
pp. 541–562, 1992.
Meade, D. B., May, M. S. J., Cheung, C-K., and Keough, G. E., Getting Started with Maple, 3rd
ed., Wiley, Hoboken, NJ, 2009.
Mickens, R.E., Nonstandard Finite Difference Models of Differential Equations, World Scientific,
Singapore, 1994.
Mikhlin, S. G., Variational Methods in Mathematical Physics [in Russian], Nauka, Moscow, 1970.
1418
Meyer-Spasche, R., Difference schemes of optimum degree of implicitness for a family of simple
ODEs with blow-up solutions, J. Comput. and Appl. Mathematics, Vol. 97, pp. 137–152, 1998.
Mikhlin, S. G. and Smolitskii, Kh. L., Approximate Solution Methods for Differential and Integral
Equations [in Russian], Nauka, Moscow, 1965.
Miles, J. W., Integral Transforms in Applied Mathematics, Cambridge University Press, Cambridge,
1971.
Milne, E., Clarkson, P. A., and Bassom, A. P., Bäcklund transformations and solution hierarchies
for the third Painlevé equation., Stud. Appl. Math., Vol. 98, No. 2, pp. 139–194, 1997.
Moriguti, S., Okuno, C., Suekane, R., Iri, M., and Takeuchi, K., Ikiteiru Suugaku — Suuri
Kougaku no Hatten [in Japanese], Baifukan, Tokyo, 1979.
Moussiaux, A., CONVODE: Un Programme REDUCE pour la Résolution des Équations Différentielles, Didier Hatier, Bruxelles, 1996.
Murdock, J. A., Perturbations. Theory and Methods, John Wiley & Sons, New York, 1991.
Murphy, G. M., Ordinary Differential Equations and Their Solutions, D. Van Nostrand, New York,
1960.
Nayfeh, A. H., Perturbation Methods, John Wiley & Sons, New York, 1973.
Nayfeh, A. H., Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.
Nikiforov, A. F. and Uvarov, V. B., Special Functions of Mathematical Physics. A Unified
Introduction with Applications, Birkhäuser Verlag, Basel–Boston, 1988.
Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer, New York, 1973.
Ockendon, J. R. and Taylor, A. B., The dynamics of a current collection system for an electric
locomotive, Proc. Royal Society of London A, Vol. 322, pp. 447–468, 1971.
Olver, F. W. J., Asymptotics and Special Functions, Academic Press, New York, 1974.
Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W. (Editors), NIST Handbook of
Mathematical Functions, NIST and Cambridge Univ. Press, Cambridge, 2010.
Olver, P. J., Application of Lie Groups to Differential Equations, Springer, New York, 1986.
Olver, P. J., Equivalence, Invariants, and Symmetry, Cambridge Univ. Press, Cambridge, 1995.
Olver, P. J., Nonlinear Ordinary Differential Equations, 2012 http://www-users.math.umn.edu/
∼olver/am /odz.pdf
Ovsiannikov, L. V., Group Analysis of Differential Equations, Academic Press, New York, 1982.
Painlevé, P., Mémoire sur les équations differentielles dont l‘integrale générale est uniforme, Bull.
Soc. Math. France, Vol. 28, pp. 201–261, 1900.
Paris, R. B., Asymptotic of High Order Differential Equations, Pitman, London, 1986.
Parkes, E. J., Observations on the tanh-coth expansion method for finding solutions to nonlinear
evolution equations, Appl. Math. Comp., Vol. 217, No. 4, pp. 1749–1754, 2010.
Pavlovskii, Yu. N. and Yakovenko, G. N., Groups admitted by dynamical systems [in Russian].
In: Optimization Methods and Applications, Nauka, Novosibirsk, pp. 155–189, 1982.
Petrovskii, I. G., Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka,
Moscow, 1970.
Petzold, L. R., Automatic selection of methods for solving stiff and nonstiff systems of ordinary
differential equations, SIAM J. Sci. Stat. Comput., Vol. 4, pp. 136–148, 1983.
Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists,
Chapman & Hall/CRC Press, Boca Raton, 2002.
Polyanin, A. D., Systems of Ordinary Differential Equations, From Website EqWorld—The World
of Mathematical Equations, 2006; http://eqworld.ipmnet.ru/en/solutions/sysode.htm.
Polyanin, A. D., Elementary theory of using invariants for solving mathematical equations, Vestnik
Samar. Gos. Univ., Estestvennonauchn. Ser. [in Russian], No. 6(65), pp. 152–176, 2008.
1419
Polyanin, A. D., Overdetermined systems of nonlinear ordinary differential equations with parameters and their applications, Bulletin of the National Research Nuclear University MEPhI [in
Russian], Vol. 5, No. 2, pp. 122–136, 2016.
Polyanin, A. D. and Chernoutsan, A. I. (Eds.) A Concise Handbook of Mathematics, Physics, and
Engineering Sciences, CRC Press, Boca Raton, 2011.
Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations, 2nd Edition, Chapman &
Hall/CRC Press, Boca Raton, 2008.
Polyanin, A. D. and Manzhirov, A. V., Handbook of Mathematics for Engineers and Scientists,
Chapman & Hall/CRC Press, Boca Raton, 2007.
Polyanin, A. D. and Shingareva, I. K., Nonlinear blow-up problems: Numerical integration
based on differential and nonlocal transformations, Bulletin of the National Research Nuclear
University MEPhI [in Russian], Vol. 6, No. 4, pp. 282–297, 2017a.
Polyanin, A. D. and Shingareva, I. K., The use of differential and non-local transformations for
numerical integration of non-linear blow-up problems, Int. J. Non-Linear Mechanics, Vol. 95,
pp. 178–184, 2017b.
Polyanin, A. D. and Shingareva, I. K., Numerical integration of blow-up problems on the basis of
non-local transformations and differential constraints, arXiv:1707.03493 [math.NA], 2017c.
Polyanin, A. D. and Shingareva, I. K., Nonlinear problems with non-monotonic blow-up solutions:
The method of non-local transformations, test problems, and numerical integration, Bulletin of
the National Research Nuclear University MEPhI [in Russian], Vol. 6, No. 5, pp. 405–424,
2017d.
Polyanin, A. D. and Shingareva, I. K., Non-monotonic blow-up problems: Test problems with
solutions in elementary functions, numerical integration based on non-local transformations,
Applied Mathematics Letters, Vol. 76, pp. 123–129, 2018.
Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential
Equations, CRC Press, Boca Raton, 1995.
Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential
Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.
Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Mathematical Physics Equations
[in Russian], Fizmatlit, Moscow, 2002.
Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations,
Chapman & Hall/CRC Press, Boca Raton, 2004.
Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, 2nd
Edition, CRC Press, Boca Raton, 2012.
Polyanin, A. D., Zaitsev, V. F., and Moussiaux, A., Handbook of First Order Partial Differential
Equations, Taylor & Francis, London, 2002.
Polyanin, A. D. and Zhurov, A. I., Parametrically defined nonlinear differential equations and their
applications in boundary layer theory, Bulletin of the National Research Nuclear University
MEPhI [in Russian], Vol. 5, No. 1, pp. 23–31, 2016a.
Polyanin, A. D. and Zhurov, A. I., Parametrically defined nonlinear differential equations and their
solutions: Application in fluid dynamics, Appl. Math. Lett., Vol. 55, pp. 72–80, 2016b.
Polyanin, A. D. and Zhurov, A. I., Functional and generalized separable solutions to unsteady
Navier–Stokes equations, Int. J. Non-Linear Mechanics, Vol. 79, pp. 88–98, 2016c.
Polyanin, A. D. and Zhurov, A. I., Parametrically defined nonlinear differential equations,
differential-algebraic equations, and implicit ODEs: Transformations, general solutions, and
integration methods, Appl. Math. Lett., Vol. 64, pp. 59–66, 2017a.
Polyanin, A. D. and Zhurov, A. I., Parametrically defined differential equations, Journal of
Physics: IOP Conf. Series, Vol. 788, 2017b, 012078; http://iopscience.iop.org/article/10.1088/
1742-6596/788/1/012028/pdf.
1420
Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 1, Elementary
Functions, Gordon & Breach Sci. Publ., New York, 1986.
Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 4, Direct
Laplace Transform, Gordon & Breach, New York, 1992a.
Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 5, Inverse
Laplace Transform, Gordon & Breach, New York, 1992b.
Puu, T., Attractions, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, SpringerVerlag, New York, 2000.
Rabier, P. J. and Rheinboldt, W. C., Theoretical and numerical analysis of differential-algebraic
equations, Handbook of Numerical Analysis, Elsevier, Vol. 8, pp. 183–540, North-Holland,
Amsterdam, 2002.
Ray J. R. and Reid, J. L., More exact invariants for the time dependent harmonic oscillator, Phys.
Letters, Vol. 71, pp. 317–319, 1979.
Reid, W. T., Riccati Differential Equations, Academic Press, New York, 1972.
Reiss, E. L., Bifurcation Theory and Nonlinear Eigenvalue Problems., W. A. Benjamin Publ., New
York, 1969.
Reiss, E. L., Column buckling: An elementary example of bifurcation. In: Bifurcation Theory and
Nonlinear Eigenvalue Problems (J. B. Keller and S. Antman, eds.), pp. 1–16, W. A. Benjamin
Publ., New York, 1969.
Reiss, E. L. and Matkowsky, B. J., Nonlinear dynamic buckling of a compressed elastic column,
Quart. Appl. Math., Vol. 29, 245–260, 1971.
Richards, D., Advanced Mathematical Methods with Maple, Cambridge University Press, Cambridge, 2002.
Ronveaux, A. (Editor), Heun’s Differential Equations, Oxford University Press, Oxford, 1995.
Rosen, G., Alternative integration procedure for scale-invariant ordinary differential equations, Intl.
J. Math. & Math. Sci., Vol. 2, pp. 143–145, 1979.
Rosenbrock, H. H., Some general implicit processes for the numerical solution of differential
equations, Comput. J., 1963, Vol. 5, No. 4, pp. 329–330.
Ross, C. C., Differential Equations: An Introduction with Mathematica, Springer, New York, 1995.
Russell, R. D. and Shampine, L. F., Numerische Mathematik, Bd. 19, No. 1, S. 1–28, 1972.
Sachdev, P. L., Nonlinear Ordinary Differential Equations and Their Applications, Marcel Dekker,
New York, 1991.
Sagdeev, R. Z., Usikov, D. A., and Zaslavsky, G. M., Nonlinear Physics: From the Pendulum to
Turbulence and Chaos, Harwood Academic Publ., New York, 1988.
Saigo, M. and Kilbas, A. A., Solution of one class of linear differential equations in terms of
Mittag-Leffler type functions [in Russian], Dif. Uravneniya, Vol. 38, No. 2, pp. 168–176, 2000.
Sanz-Serna, J. M. and Calvo, M. P., Numerical Hamiltonian Problems: Applied Mathematics and
Mathematical Computation, Chapman & Hall, London, 1994.
Schiesser, W. E., Computational Mathematics in Engineering and Applied Science: ODEs, DAEs,
and PDEs, CRC Press, Boca Raton, 1994.
Shampine, L. F. and M. K. Gordon, M. K., Computer Solution of Ordinary Differential Equations:
the Initial Value Problem, W. H. Freeman, San Francisco, 1975.
Shampine, L. F. and Watts, H. A., The art of writing a Runge–Kutta code I. In: Mathematical
Software III (J. R. Rice, editor), Academic Press, New York, 1977.
Shampine, L. F. and Watts, H. A., The art of writing a Runge–Kutta code II, Appl. Math. Comput.,
Vol. 5, pp. 93–121, 1979.
1421
Shampine, L. F. and Gear, C. W., A user’s view of solving stiff ordinary differential equations,
SIAM Review, Vol. 21, pp.1–17, 1979.
Shampine, L. F. and Baca, L. S., Smoothing the extrapolated midpoint rule, Numer. Math., Vol. 41,
pp. 165–175, 1983.
Shampine, L. F., Control of step size and order in extrapolation codes, J. Comp. Appl. Math.,
Vol. 18, pp. 3–16, 1987.
Shampine, L. F., Numerical Solution of Ordinary Differential Equations, Chapman & Hall/CRC
Press, Boca Raton, 1994.
Shampine, L. F. and Reichelt, M. W., The MATLAB ODE Suite, SIAM Journal on Scientific
Computing, Vol. 18, pp. 1–22, 1997.
Shampine, L. F., Reichelt, M. W., and Kierzenka, J., Solving index-1 DAEs in MATLAB and
Simulink, SIAM Rev., Vol. 41, No. 3, pp. 538552, 1999.
Shampine, L. F. and Corless, R. M., Initial value problems for ODEs in problem solving
environments, Journal of Computational and Applied Mathematics, Vol. 125, No. 1–2, pp.31–
40, 2000.
Shampine, L. F. and Thompson, S., Solving DDEs in MATLAB, Appl. Numer. Math., Vol. 37,
pp. 441–458, 2001.
Shampine, L. F., Gladwell, I., and Thompson, S., Solving ODEs with MATLAB, Cambridge
University Press, Cambridge, UK, 2003.
Shingareva, I. K., Investigation of Standing Surface Waves in a Fluid of Finite Depth by Computer
Algebra Methods, PhD thesis, Institute for Problems in Mechanics, Russian Academy of
Sciences, Moscow, 1995.
Shingareva, I. K. and Lizárraga-Celaya, C., Maple and Mathematica. A Problem Solving
Approach for Mathematics, 2nd ed., Springer, Wien, New York, 2009.
Shingareva, I. K. and Lizárraga-Celaya, C., Solving Nonlinear Partial Differential Equations
with Maple and Mathematica, Springer, Wien, New York, 2011.
Shingareva, I. K. and Lizárraga-Celaya, C., On different symbolic notations for derivatives, The
Mathematical Intelligencer, Vol. 37, No. 3, pp. 33–38, 2015.
Simmons, G. F., Differential Equations with Applications and Historical Notes, McGraw-Hill, New
York, 1972.
Sintsov, D. M., Integration of Ordinary Differential Equations [in Russian], Kharkov, 1913.
Slavyanov, S. Yu., Lay, W., and Seeger, A., Heun’s Differential Equation, University Press,
Oxford, 1955.
Sofroniou, M. and Spaletta, G., Construction of explicit Runge–Kutta pairs with stiffness
detection, Mathematical and Computer Modelling (Special Issue on the Numerical Analysis
of Ordinary Differential Equations), Vol. 40, No. 11–12, pp. 1157–1169, 2004.
Sofroniou, M. and Spaletta, G., Derivation of symmetric composition constants for symmetric
integrators, Optimization Methods and Software, Vol. 20, No. 4–5, pp. 597–613, 2005.
Sofroniou, M. and Spaletta, G., Hybrid solvers for splitting and composition methods, J. Comp.
Appl. Math. (Special Issue from the International Workshop on the Technological Aspects of
Mathematics), Vol. 185, No. 2, pp. 278–291, 2006.
Stepanov, V. V., A Course of Differential Equations, 7th Edition [in Russian], Gostekhizdat,
Moscow, 1958.
Sparrow, C., The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New
York, 1982.
Stephani, H., Differential Equations: Their Solution Using Symmetries (edited by M. A. H.
MacCallum), Cambridge University Press, New York, 1989.
1422
Strang, G., On the construction of difference schemes, SIAM J. Num. Anal., Vol. 5, pp. 506–517,
1968.
Stuart, M. and Floater, M. S., On the computation of blow-up, European J. Applied Math., Vol. 1,
No. 1, pp. 47–71, 1990.
Stephani, H., Differential Equations: Their Solutions Using Symmetries, Cambridge University
Press, Cambridge, 1989.
Strang, G., On the construction of difference schemes, SIAM J. Num. Anal., Vol. 5, pp. 506–517,
1968.
Svirshchevskii, S. R., Lie–Bäcklund symmetries of linear ODEs and generalized separation of
variables in nonlinear equations, Phys. Letters A, Vol. 199, pp. 344–348, 1995.
Takayasu, A., Matsue, K., Sasaki, T., Tanaka, K., Mizuguchi, M., and Oishi, S., Numerical validation of blow-up solutions of ordinary differential equations, J. Comput. Applied Mathematics,
Vol. 314, pp. 10–29, 2017.
Temme, N. M., Special Functions. An Introduction to the Classical Functions of Mathematical
Physics, Wiley-Interscience, New York, 1996.
Tenenbaum, M. and Pollard, H., Ordinary Differential Equations, Dover Publications, New York,
1985.
Teodorescu, P. P., Mechanical Systems, Classical Models: Volume II: Mechanics of Discrete and
Continuous Systems, Springer, Berlin, 2009.
Tikhonov, A. N., Vasil’eva, A. B., and Sveshnikov, A. G., Differential Equations [in Russian],
Nauka, Moscow, 1985.
Trotter, H. F., On the product of semi-group operators, Proc. Am. Math. Soc., Vol. 10, pp. 545–551,
1959.
Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
Van Hulzen, J. A. and Calmet, J., Computer algebra systems. In: Computer Algebra, Symbolic and
Algebraic Manipulation, (Buchberger, B., Collins, G. E., and Loos, R., eds.), 2nd ed., pp. 221–
243, Springer, Berlin, 1983.
Vasil’eva, A. B. and Nefedov, H. H., Nonlinear Boundary Value Problems [in Russian], Lomonosov
MSU, Moscow, 2006; http://math.phys.msu.ru/data/57/Nefmaterial.pdf.
Verner, J. H., Explicit Runge–Kutta methods with estimates of the local truncation error, SIAM
Journal of Numerical Analysis, Vol. 15, No. 4, pp. 772–790, 1978.
Vinogradov, I. M. (Editor), Mathematical Encyclopedia [in Russian], Soviet Encyclopedia, Moscow,
1979.
Vinokurov, V. A. and Sadovnichii, V. A., Arbitrary-order asymptotic relations for eigenvalues and
eigenfunctions in the Sturm–Liouville boundary-value problem on an interval with summable
potential [in Russian], Izv. RAN, Ser. Matematicheskaya, Vol. 64, No. 4, pp. 47–108, 2000.
Vitanov, N. K. and Dimitrova, Z. I., Application of the method of simplest equation for obtaining
exact traveling-wave solutions for two classes of model PDEs from ecology and population
dynamics, Commun. Nonlinear Sci. Numer. Simulat., Vol. 15, No. 10, pp. 2836–2845, 2010.
Vvedensky, D. D., Partial Differential Equations with Mathematica, Addison-Wesley, Wokingham,
1993.
Wang, D.-S., Ren, Y.-J., and Zhang, H.-Q., Further extended sinh-cosh and sin-cos methods and
new non traveling wave solutions of the (2+1)-dimensional dispersive long wave equations,
Appl. Math. E-Notes, Vol. 5, pp. 157–163, 2005.
Wang, M. L., Li, X., Zhang, J., The G′/G-expansion method and evolution equations in mathematical physics, Phys. Lett. A, Vol. 372, pp. 417–421, 2008.
Wang, S.-H., On S-shaped bifurcation curves, Nonlinear Anal., Vol. 22, No. 12, pp. 1475–1485,
1994.
1423
Wang, S.-H., On the evolution and qualitative behaviors of bifurcation curves for a boundary value
problem, Nonlinear Analysis, Vol. 67, pp. 1316–1328, 2007.
Wasov, W., Asymptotic Expansions for Ordinary Differential Equations, John Wiley & Sons, New
York, 1965.
Wazwaz, A. M., A sine-cosine method for handling nonlinear wave equations, Math. and Computer
Modelling, Vol. 40, No. 5-6, pp. 499–508, 2004.
Wazwaz, A. M., The tanh-coth method for solitons and kink solutions for nonlinear parabolic
equations, Appl. Math. Comput., Vol. 188, No. 2, pp. 1467–1475, 2007a.
Wazwaz, A. M., Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by
the tanh-coth method, Appl. Math. Comput., Vol. 190, No. 1, pp. 633–640, 2007b.
Wazwaz, A. M., Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh-coth method and Exp-function method, Appl. Math. Comput.,
Vol. 202, No. 1, pp. 275–286, 2008.
Wazzan, L., A modified tanh-coth method for solving the KdV and the KdV–Burgers’ equations,
Commun. Nonlinear Sci. and Numer. Simulation, Vol. 14, No. 2, pp. 443–450, 2009.
Wester, M. J., Computer Algebra Systems: A Practical Guide, Wiley, Chichester, UK, 1999.
Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, Vols. 1–2, Cambridge
University Press, Cambridge, 1952.
Wiens, E. G., Bifurcations and Two Dimensional Flows. In: Egwald Mathematics: http://www.
egwald.ca/nonlineardynamics/bifurcations.php
Wiggins, S., Global bifurcations and Chaos: Analytical Methods, Springer-Verlag, New York, 1988.
Wolfram, S., A New Kind of Science, Wolfram Media, Champaign, IL, 2002.
Wolfram, S., The Mathematica Book, 5th ed., Wolfram Media, Champaign, IL, 2003.
Yanenko, N. N., The compatibility theory and methods of integration of systems of nonlinear
partial differential equations. In: Proceedings of All-Union Math. Congress, Vol. 2, pp. 613–
621, Nauka, Leningrad, 1964.
Yermakov, V. P., Second-order differential equations. Integrability conditions in closed form
[in Russian], Universitetskie Izvestiya, Kiev, No. 9, pp. 1–25, 1880.
Zaitsev, O. V. and Khakimova, Z. N., Classification of new solvable cases in the class of
polynomial differential equations [in Russian], Topical Issues of Modern Science [Aktual’nye
voprosy sovremennoi nauki], No. 3, pp. 3–11, 2014.
Zaitsev, V. F., Universal description of symmetries on a basis of the formal operators, Proc. Intl.
Conf. MOGRAN-2000 “Modern Group Analysis for the New Millennium,” USATU Publishers,
Ufa, pp. 157–160, 2001.
Zaitsev, V. F. and Huan, H. N., Analogues of variational symmetry of third order ODE, Izvestia:
Herzen University Journal of Humanities & Sciences, No. 154, pp. 33–41, 2013.
Zaitsev, V. F. and Huan, H. N., Analogues of variational symmetry of the equation of the type
y ′′′ = F (y, y ′ , y ′′ ), Izvestia: Herzen University Journal of Humanities & Sciences, No. 163,
pp. 7–17, 2014.
Zaitsev, V. F. and Linchuk, L. V., Differential Equations (Structural Theory), Part I [in Russian],
A. I. Herzen Russian State Pedagogical University, St. Petersburg, 2015.
Zaitsev, V. F. and Linchuk, L. V., Differential Equations (Structural Theory), Part II [in Russian],
A. I. Herzen Russian State Pedagogical University, St. Petersburg, 2009.
Zaitsev, V. F. and Linchuk, L. V., Differential Equations (Structural Theory), Part III [in Russian],
A. I. Herzen Russian State Pedagogical University, St. Petersburg, 2014.
1424
Zaitsev, V. F. and Linchuk, L. V., Six new factorizable classes of 2nd-order ODEs [in Russian],
Some Topical Problems of Modern Mathematics and Mathematical Education, Proc. LXVIII
International Conference “Herzen Readings – 2016” (11–15 April 2016, St. Petersburg, Russia),
A. I. Herzen Russian State Pedagogical University, 2016, pp. 82–89.
Zaitsev, V. F., Linchuk, L. V., and Flegontov, A. V., Differential Equations (Structural Theory),
Part IV [in Russian], A. I. Herzen Russian State Pedagogical University, St. Petersburg, 2014.
Zaitsev, V. F. and Polyanin, A. D., Discrete-Group Methods for Integrating Equations of Nonlinear
Mechanics, CRC Press, Boca Raton, 1994.
Zaitsev, V. F. and Polyanin, A. D., Handbook of Nonlinear Differential Equations: Exact Solutions
and Applications in Mechanics [in Russian], Nauka, Moscow, 1993.
Zaitsev, V. F. and Polyanin, A. D., Handbook of Ordinary Differential Equations [in Russian],
Fizmatlit, Moscow, 2001.
Zaslavsky, G. M., Introduction to Nonlinear Physics [in Russian], Fizmatlit, Moscow, 1988.
Zayed, E. M. E., The G′/G-method and its application to some nonlinear evolution equations, J.
Appl. Math. Comput., Vol. 30, pp. 89–103, 2009.
Zhang, H., New application of the G′ /G-expansion method, Commun. Nonlinear Sci. Numer.
Simulat., Vol. 14, pp. 3220–3225, 2009.
Zhang, S., Application of Exp-function method to a KdV equation with variable coefficients, Phys.
Letters A, Vol. 365, No. 5–6, pp. 448–453, 2007.
Zhuravlev, V. Ph., The solid angle theorem in rigid body dynamics, J. Appl. Math. Mech., Vol. 60,
No. 2, pp. 319–322, 1996.
Zhuravlev, V. Ph., Foundations of Theoretical Mechanics [in Russian], Fizmatlit, Moscow, 2001.
Zhuravlev, V. Ph. and Klimov, D. M., Applied Methods in Oscillation Theory [in Russian], Nauka,
Moscow, 1988.
Zimmerman, R. L. and Olness, F., Mathematica for Physicists, Addison-Wesley, Reading, MA,
1995.
Zinchenko, N. S., A Lecture Course on Electron Optics [in Russian], Kharkov State University,
Kharkov, 1958.
Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, New York, 1997.
1425
View publication stats