HANDBOOK OF EXACT SOLUTIONS FOR

HANDBOOK OF EXACT SOLUTIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

Andrei D. Polyanin Valentin F. Zaitsev

CRC PRESS, 1995 Boca Raton

New York

London

FOREWORD Exact solutions have always played and still play an important role in properly understanding the qualitative features of many phenomena and processes in various fields of natural science. Equations of applied and theoretical physics often contain parameters or functions which are found experimentally and therefore are not stringently fixed. At the same time, equations that model real phenomena and processes must be sufficiently simple to make possible their analysis and solution. It is natural to adopt, as one of feasible criterions of simplicity, the requirement that the model equation admits a solution in a closed form. It should be noted that even exact solutions of nonlinear equations (including those without a clear physical sense and which do not correspond to real phenomena and processes) play an important role of “test” problems for verifying the correctness and assessment of accuracy of various numerical, asymptotic, and approximate methods. Moreover, the model equations and problems admitting exact solutions serve as the basis for the development of new numerical, asymptotic, and approximate methods, which, in turn, enable us to study more complicated problems having no analytical solution. This book contains nearly 5000 ordinary differential equations and their solutions. The table below compares data presented in this work with those of currently available handbooks concerning the general number of concrete second- and higher-order ordinary differential equations analyzed. The order of equations

E. Kamke (1976)

M. Murphy (1960)

This book

Second order Third order

249 13

315 22

1227 587

Fourth order Higher order

3 3

3 9

75 160

268

349

2049

Total number of equations

When selecting the material, the authors gave preference to the following two types of equations: 1. Equations that traditionally attracted the attention of many researchers: those of the simplest appearance but involving the most difficulties for integration (Abel equations, Emden—Fowler equations, Painlevé equations, etc.). 2. Equations that encountered in various applications (in the theory of heat and mass transfer, nonlinear mechanics, hydrodynamics, the theory of nonlinear oscillations, the theory of combustion, chemical engineering science, etc.). Special attention is paid to equations containing arbitrary functions. The other equations contain one or more arbitrary parameters (i.e., actually, this book deals with whole families of ordinary differential equations) which can be fixed by a reader at will. Many solutions have been obtained just recently with the aid of new (discrete group) methods described in other books by the authors (1993, 1994). When compiling this book, the handbooks by E. Kamke (1976), M. Murphy (1960), and D. Zwillinger (1989) were partly used in which one can find basic notions and definitions of the theory of ordinary differential equations, apart from concrete equations. In these handbooks, classical and some new methods of solving differential equations are described as well—see also the books by E.L. Ince (1964), P.J. Olver (1986), and N.H. Ibragimov

(1993). The latter books give a great number of references to the original papers and books by other authors, which are devoted to exact solutions and methods of the theory of ordinary differential equations. In addition, when describing solutions of linear ordinary differential equations, which are connected to higher transcendental functions (Bessel, Legendre, Mathieu, hypergeometric, etc.), the handbooks by G. Beitmen and A. Erdeii (1953–1955), M. Abramowitz and I.A. Stegun (1964) were used. In some sections of this book, asymptotic solutions of some classical equations of nonlinear mechanics and theoretical physics are also given, which are discussed in the books by J.D. Cole (1968), M.V. Fedoryuk (1983), and A.H. Nayfeh (1973, 1971) in detail. The detailed table of contents enables a reader to quickly navigate through this book in searching for desired equations. The authors hope that this book will be helpful for a wide range of scientists, lectures, engineers, and students engaged in the fields of mathematics, physics, mechanics, and chemical engineering science. Andrei D. Polyanin Valentin F. Zaitsev

Some Remarks and Notation 1. In this book, in the original equations the independent variable is denoted by x, and the dependent one is denoted by y. In the given solutions, the symbols C, C0 , C1 , C2 , . . . stand for arbitrary integration constants. 2. The following notation is used for derivatives: yx′ = ′′′′ yxxxx =

d4 y dn y , and yx(n) = 4 dx dxn

d2 y dy d3 y ′′ ′′′ , yxx = , y = , xxx dx dx2 dx3

with n ≥ 5.

( d )n 3. In some cases, we use the operator notation f g which is defined by the dx recurrence relation [ ] ( d ( d )n d )n−1 g(x) = f (x) g(x) . f (x) f (x) dx dx dx 4. In some sections of the book (see, for example, 1.3, 2.3–2.6, 3.2–3.4), for the sake of brevity, solutions are represented as several formulae containing the terms with the signs “±” and “∓.” By this is meant two formulae—one correspond to the upper signs, and another corresponds to the lower signs. For example, the solution of equation 1.3.1.6 can be written in the parametric form ∫ [ ] −1 2 −1 2 x = af exp(∓τ ), y = af exp(∓τ )±2τ f , where f = exp(∓τ 2 ) dτ −C, A = ∓2a2 . This is equivalent to that the solutions of equation 1.3.1.6 are given by the formulae ∫ [ ] x = af −1 exp(−τ 2 ), y = af −1 exp(−τ 2 )+2τ f , where f = exp(−τ 2 ) dτ −C, A = −2a2 and x = af

−1

2

exp(τ ), y = af

−1

[ ] exp(τ 2 ) − 2τ f ,

∫ where f =

exp(τ 2 ) dτ − C, A = 2a2 .

5. When referencing to a particular equation, the notation like “4.1.2.5” stands for “equation 5 in Subsection 4.1.2.” 6. The book includes two supplements that provide a reader with useful information on some elementary and special functions which appear in solutions of the differential equation outlined. 7. References that may be helpful for a reader are given at the end of the book.

THE AUTHORS Andrei D. Polyanin, Ph.D., D.Sc., is a noted scientist in the fields of ordinary differential equations, engineering and applied mathematics, heat and mass transfer, nonlinear mechanics, and chemical engineering science. Professor Polyanin graduated from the Faculty of Mechanics and Mathematics of the Moscow State University in 1974 and received his Candidate of Sciences (Ph.D.) degree in 1981 (at the Institute for Problems in Mechanics of the U.S.S.R. Academy of Sciences). His Ph.D. thesis was devoted to the asymptotic analysis of the problems of heat and mass transfer. In 1986, Professor Polyanin received his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the mass and heat exchange between reacting particles and flow. Since 1975, Professor Polyanin is a member of staff of the Institute for Problems in Mechanics of the Russian Academy of Sciences. Professor Polyanin has made important contributions to new approximate analytical methods in the theory of heat and mass transfer, hydrodynamics, and chemical engineering science, as well as to new methods of the theory of ordinary differential equations. In 1991 he was awarded a Chaplygin Prize of the U.S.S.R. Academy of Sciences for his research in mechanics. Professor Polyanin has published more than 100 research papers and seven books. He is also an author of three patents. Valentin F. Zaitsev, Ph.D., D.Sc., is a noted scientist in the fields of ordinary differential equations, mathematical physics, and nonlinear mechanics. Professor Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Politechnical Institute (now Saint-Petersburg Technical University) in 1969 and received his Candidate of Sciences (Ph.D.) degree in 1983 (at the Leningrad State University). His Ph.D. thesis was devoted to the group approach to the study of some classes of ordinary differential equations. In 1992, Professor Zaitsev received his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the discrete-group analysis of the ordinary differential equations. Since 1971, Professor Zaitsev is in the Research Institute for Computational Mathematics and Control Processes of the St.-Petersburg State University. He is also a Professor at the Russian State Pedagogical University (St.-Petersburg), the Orel State Pedagogical Institute, and Orel State Politechnical Institute. Professor Zaitsev has made important contributions to new methods in the theory of ordinary differential equations. He is an author of more than 70 scientific publications, including five monographs and the patent. Zaitsev V.F. also read the theoretical course at the Leningrad Conservatory, later participating in development of mathematical methods in musical sciences.

CONTENTS Annotation Foreword

.............................................................

iii

...............................................................

v

Some Remarks and Notation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. Simplest Equations with Arbitrary Functions Integrable in a Closed Form . . . 1.1.1. Equations of the Form yx′ = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Equations of the Form yx′ = f (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3. Separable Equations yx′ = f (x)g(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4. Linear Equation g(x)yx′ = f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5. Bernoulli Equation g(x)yx′ = f1 (x)y + fn (x)y n . . . . . . . . . . . . . . . . . . . . 1.1.6. Homogeneous Equation yx′ = f (y/x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 2 2

1.2. Riccati Equations: g(x)yx′ = f2 (x)y 2 + f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . 1.2.1. Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 1.2.5. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 1.2.6. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1.2.7. Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 1.2.8. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9. Some transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 4 12 15 16 18 22 23 27

1.3. Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Equations of the Form yyx′ − y = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Equations of the Form yyx′ = f (x)y + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3. Equations of the Form yyx′ = f1 (x)y + f0 (x) . . . . . . . . . . . . . . . . . . . . . . 1.3.4. Equations of the Form [g1 (x)y + g0 (x)]yx′ = f2 (x)y 2 + f1 (x)y + f0 (x) . 1.3.5. Some Types of First and Second Order Equations Reducible to Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 45 46 50 55

1.4. Equations Containing Polynomial Functions of y . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.4.1. Abel Equations of the First Kind yx′ = f3 (x)y 3 +f2 (x)y 2 +f1 (x)y +f0 (x) 57 1.4.2. Equations of the Form (A22 y 2 + A12 xy + A11 x2 + A0 )yx′ = B22 y 2 + B12 xy + B11 x2 + B0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.4.3. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.4.4. 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 . . . . . . . . . . . . . . . . . . . . . . . 73 1.5. Nonlinear Equations of the Form f (x, y)yx′ = g(x, y) Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 1.5.3. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 1.5.4. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 1.5.5. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .

78 78 82 87 89 90 92

1.6. Equations Not Solved for Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1.6.1. Equations of the Second Degree in yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1.6.2. Equations of the Third Degree in yx′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 k 1.6.3. Equations of the Form (yx′ ) = f (y) + g(x) . . . . . . . . . . . . . . . . . . . . . . . 104 1.6.4. Other equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1.7. Equations of the Form F (x, y)yx′ = G(x, y) Containing Arbitrary Functions . . 1.7.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2. Equations Containing Exponential and Hyperbolic Functions . . . . . . . . 1.7.3. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 1.7.4. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 116 118 120 121 123

F (x, y, yx′ )

1.8. Equations of the Form = 0 Not Solved for the Derivative and Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 1.8.1. Some Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 1.8.2. Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2. Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 2.1.6. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 2.1.7. Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 2.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.1.10. Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.11. Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.12. Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 130 166 173 178 181 195

′′ 2.2. Autonomous Equations yxx = F (y, yx′ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.2.1. Equations of the Form yxx − yx′ = f (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.2.2. Equations of the Form yxx + f (y)yx′ + y = 0 . . . . . . . . . . . . . . . . . . . . . . ′′ 2.2.3. Lienard Equations yxx + f (y)yx′ + g(y) = 0 . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.2.4. Rayleigh Equations yxx + f (yx′ ) + g(y) = 0 . . . . . . . . . . . . . . . . . . . . . . . .

227 228 232 235 238

′′ 2.3. Emden—Fowler Equation yxx = Axn y m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. First Integrals (Conservation Laws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 247 250

202 212 220 222 226

′′ 2.4. Equations of the Form yxx = A1 xn1 y m1 + A2 xn2 y m2 . . . . . . . . . . . . . . . . . . . . . . 250 2.4.1. Classification Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 2.4.2. Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 ′′ 2.5. Generalized Emden—Fowler Equation yxx = Axn y m (yx′ ) . . . . . . . . . . . . . . . . . . 2.5.1. Classification Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3. Some Formulae and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . l

277 277 282 299

′′ 2.6. Equations of the Form yxx = A1 xn1 y m1 (yx′ ) 1 + A2 xn2 y m2 (yx′ ) 2 . . . . . . . . . . . . ′′ = A1 x−1 yx′ + A2 xn y m . . . . . . . 2.6.1. Modified Emden—Fowler Equation yxx l ′′ n1 m1 2.6.2. Equations of the Form yxx = (A1 x y + A2 xn2 y m2 )(yx′ ) . . . . . . . . . l l−1 ′′ 2.6.3. Equations of the Form yxx = σAxn y m (yx′ ) + Axn−1 y m+1 (yx′ ) .... 2.6.4. Other Equations (l1 ̸= l2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301 301 313 348 367

′′ 2.7. Equations of the Form yxx = f (x)g(y)h(yx′ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.7.1. Equations of the Form yxx = f (x)g(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2. Equations Containing Power Functions (h ̸≡ const) . . . . . . . . . . . . . . . . 2.7.3. Equations Containing Exponential Functions (h ̸≡ const) . . . . . . . . . . . 2.7.4. Equations Containing Hyperbolic Functions (h ̸≡ const) . . . . . . . . . . . . 2.7.5. Equations Containing Trigonometric Functions (h ̸≡ const) . . . . . . . . . 2.7.6. Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

372 373 376 380 384 386 387

2.8. Some Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 2.8.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2. Painlevé Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3. Equation Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 2.8.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 2.8.5. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 2.8.6. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . .

388 388 394 400 407 410 412

2.9. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.9.1. Equations of the Form F (x, y)yxx + G(x, y) = 0 . . . . . . . . . . . . . . . . . . . ′′ + G(x, y)yx′ + H(x, y) = 0 . . . . . . . . 2.9.2. Equations of the Form F (x, y)yxx ∑M m ′′ + m=0 Gm (x, y)(yx′ ) = 0 2.9.3. Equations of the Form F (x, y)yxx (M = 2, 3, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′ 2.9.4. Equations of the Form F (x, y, yx′ )yxx + G(x, y, yx′ ) = 0 . . . . . . . . . . . . . ′ ′′ 2.9.5. Equations of the Form F (x, y, yx , yxx ) = 0 . . . . . . . . . . . . . . . . . . . . . . . . 2.9.6. General Equations Admitting the Order Reduction . . . . . . . . . . . . . . . . 2.9.7. Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

420 420 423

l

l

417

427 430 438 440 444

3. Third Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 3.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 3.1.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 3.1.6. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 3.1.7. Equations Containing Inverse Trigonometric Functions . . . . . . . . . . . . . 3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.9. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .

449 449 450 471 477 487 490 503

′′′ ′′ 3.2. Equations of the Form yxxx = Axα y β (yx′ ) (yxx ) ......................... 3.2.1. Preliminary Comments. Classification Table . . . . . . . . . . . . . . . . . . . . . . ′′′ 3.2.2. Equations of the Form yxxx = Ay β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ′′′ 3.2.3. Equations of the Form yxxx = Axα y β . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Equations with |γ| + |δ| ̸= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Some Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

528 528 535 537 538 572

γ

δ

510 518

′′′ ′′ 3.3. Equations of the Form yxxx = f (y)g(yx′ )h(yxx ) ........................... 3.3.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

573 573 577 581

3.4. Some Nonlinear Equations with Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 3.4.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 3.4.3. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 3.4.5. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . .

584 584 589 591 594 596

3.5. Nonlinear Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . ′′′ 3.5.1. Equations of the Form F (x, y)yxxx + G(x, y) = 0 . . . . . . . . . . . . . . . . . . ′′′ 3.5.2. Equations of the Form F (x, y, yx′ )yxxx + G(x, y, yx′ ) = 0 . . . . . . . . . . . . . ′ ′′′ ′ ′′ 3.5.3. Equations of the Form F (x, y, yx )yxxx + G(x, H(x, y, yx′ ) = 0 ∑ y, yx )yxx + ′ ′′′ ′ ′′ α 3.5.4. Equations of the Form F (x, y, yx )yxxx + α Gα (x, y, yx )(yxx ) =0 .. ′ ′′ ′′′ ′ ′′ 3.5.5. Equations of the Form F (x, y, yx , yxx )yxxx + G(x, y, yx , yxx ) = 0 . . . . .

599 599 601 606 611 613

4. Fourth Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 4.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Equations Containing Exponential, Hyperbolic, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Equation Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 4.1.5. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6. Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

617 617 617

4.2. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Equation Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Equations Containing Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .

633 633

625 627 629 633

636 638

5. Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 5.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 5.1.5. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6. Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

645 645 646 652 654 656 661

5.2. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Equations Containing Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Equations Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 5.2.4. Equations Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 5.2.5. Equations Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 5.2.6. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . .

662 662 666 667 669 670 672

Supplement 1. Some Elementary Functions and Their Properties . . . . . . . . 681 1.1. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Relations Between Trigonometric Functions of Identical Argument . . . 1.1.3. Reduction Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4. Addition Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5. Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . 1.1.6. Product of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7. Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8. Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . . 1.1.9. Euler and de Moivre Formulae, Relation to Hyperbolic Functions . . . . 1.1.10. Differentiation and Integration Formulae . . . . . . . . . . . . . . . . . . . . . . . . .

681 681 681 681 682 682 682 682 683 683 684

1.2. Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Relations Between Hyperbolic Functions of Identical Argument . . . . . . 1.2.3. Addition Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Addition and Subtraction of Hyperbolic Functions . . . . . . . . . . . . . . . . . 1.2.5. Product of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6. Powers of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7. Hyperbolic Functions of Multiple Argument . . . . . . . . . . . . . . . . . . . . . . . 1.2.8. Relation to Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9. Differentiation and Integration Formulae . . . . . . . . . . . . . . . . . . . . . . . . .

684 684 685 685 685 685 686 686 686 687

1.3. Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Relation Between Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 1.3.3. Addition and Subtraction of Inverse Trigonometric Functions . . . . . . . 1.3.4. Differentiation and Integration Formulae . . . . . . . . . . . . . . . . . . . . . . . . .

687 687 688 689 689

1.4. Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Relation to Logarithmic Functions and Simplest Relations . . . . . . . . . . 1.4.2. Relations Between Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . 1.4.3. Addition and Subtraction of Inverse Hyperbolic Functions . . . . . . . . . . 1.4.4. Differentiation and Integration Formulae . . . . . . . . . . . . . . . . . . . . . . . . .

689 689 689 690 690

1.5. Some Conventional Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2. Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3. Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

690 690 690 691

Supplement 2. Some Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 2.1. Gamma-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 2.2. Bessel functions Jν and Yν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Basic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Bessel functions for ν = ±n ± 12 ; n = 0, 1, 2, . . . . . . . . . . . . . . . . . . . . 2.2.3. Wronskians an Similar Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. Integrals with Bessel Functions on Closed Intervals . . . . . . . . . . . . . . . . 2.2.6. Asymptotic Expansion, as |x| → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

694 694 694 695 695 695 695

2.3. Modified Bessel Functions Iν and Kν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Basic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Modified Bessel Functions for ν = ±n ± 21 ; n = 0, 1, 2, . . . . . . . . . . . 2.3.3. Wronskians and Similar Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Integrals with Modified Bessel Functions on Closed Intervals . . . . . . . . 2.3.6. Asymptotic Expansion, as x → +∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

696 696 696 697 697 697 697

2.4. Degenerate Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Integrals with Degenerate Hypergeometric Functions . . . . . . . . . . . . . . . 2.4.5. Asymptotic Expansion, as |x| → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

697 697 698 699 699 699

2.5. Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Trigonometric Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3. Some Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

699 699 700 700 701

2.6. The Weierstrass function ℘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 2.6.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 2.6.2. Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 References Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

References

Abramowitz, M., and Stegun, I.A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Washington, National Bureau of Standards Applied Mathematics, 1964. Bateman, H., and Erd´ elyi, A., Higher Transcendental Functions, Vol. 1, New York, McGraw-Hill Book Comp., 1953. Bateman, H., and Erd´ elyi, A., Higher Transcendental Functions, Vol. 2, New York, McGraw-Hill Book Comp., 1953. Bateman, H., and Erd´ elyi, A., Higher Transcendental Functions, Vol. 3, New York, McGraw-Hill Book Comp., 1955. Bercovich, L.M., Factorization and Transformations of Ordinary Differential Equations, Saratov, Saratov University Publ., 1989. Bluman, G.W., and Cole, J.D. Similarity Methods for Differential Equations, New York, Springer—Verlag, 1974. Cole, J.D., Perturbation Methods in Applied Mathematics, Waltham, Blaisdell Publ. Comp., 1968. Fedoryuk, .V., Asymptotic Methods for Linear Ordinary Differential Equations, Moscow, Nauka, 1983 (in Russian). Gradshteyn, I.S., and Ryzhik, I.M. Tables of Integrals, Series, and Products, New York, Academic Press, 1980. Gromak, V.I., and Lukashevich, N.A. Analytical Properties of Solutions of Painlevé Equations, Minsk, Universitetskoye, 1990 (in Russian). Hartman, P., Ordinary Differential Equations, New York, Jonh Wiley & Sons, 1964. Hill, J.M., Solution of differential equations by means of one-parameter groups, Mass.: Pitman, Marshfield, 1982. Ibragimov, N.H., CRC Handbook of Lie Group to Differential Equations, Vol. 1, Boca Raton, CRC Press, 1993. Ince, E.L., Ordinary Differential Equations, New York, Dover Publications, 1964. Kamke, E., Handbook on Ordinary Differential Equations, Moscow, Nauka, 1976 (in Russian). Korn, G.A., and Korn, T.M., Mathematical Handbook for Scientists and Engineers, New York, McGraw-Hill Book Comp., 1961. Matveev, N.M., Methods of Integration of Ordinary Differential Equations, Moscow, Vysshaya Shkola, 1967 (in Russian). Murphy, G.M., Ordinary Differential Equations and Their Solutions, New York, D. Van Nostrand, 1960. Nayfeh, A.H., Perturbation Methods, New York, John Wiley & Sons, 1973.

Nayfeh, A.H., Introduction to Perturbation Techniques, New York, John Wiley & Sons, 1981. Olver, P.J., Application of Lie Group to Differential Equations, New York, Springer— Verlag, 1986. Ovsiannikov, L.V., Group Analysis of Differential Equations, New York, Academic Press, 1982. Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integrals and Series, Vol. 1, Elementary Functions, New York, Gordon & Breach Sci. Publ., 1986. Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integrals and Series, Vol. 2, Special Functions, New York, Gordon & Breach Sci. Publ., 1986. Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integrals and Series, Vol. 3, More Special Functions, New York, Gordon & Breach Sci. Publ., 1988. Zaitsev, V.F. and Polyanin, A.D., Discrete-Group Methods for Integrating Equations of Nonlinear Mechanics, Boca Raton, CRC Press—Begel House, 1994. Zaitsev, V.F. and Polyanin, A.D., Handbook on Nonlinear Differential Equations, Moscow, Nauka, 1993 (in Russian). Zwillinger, D., Handbook of Differential Equations, San Diego, Academic Press, 1989.

Index

A Abel equations of the first kind, 29, 57 Abel equations of the second kind, 29–57 Airy equation, 130 Airy functions, 130 arbitrary functions, 1, 116, 124, 212, 420, 518, 599, 629, 638, 656, 672 arbitrary parameters, 78, 388, 584 associated Legendre functions, 699 associated Legendre functions of the first kind, 699 associated Legendre functions of the second kind, 699 asymptotic expansion, 223, 633, 661, 695 asymptotic solutions of fourth order linear equations, 633 asymptotic solutions of n th order linear equations, 661 asymptotic solutions of second order linear equations, 222–226 autonomous equation, 614, 642, 678

B Bäcklund transformation, 387, 395 Basset function, 696 Bernoulli equation, 2 Bessel equation, 145 Bessel functions, 145, 694 Bessel functions of the first kind, 694 Bessel functions of the second kind, 694 binomial coefficients, 683, 690

C canonical form, 3, 129, 335 Clerot equation, 125 conservation laws, 247 Coulomb law, 82 cylindrical function, 145, 694

D de Moivre formulae, 683 degenerate hypergeometric equation, 137 degenerate hypergeometric functions, 697 Duffing equation, 235 dynamical system, 65

E elementary functions, 681 elliptic integral of the first kind, 558 elliptic integral of the second kind, 41, 230 Emden—Fowler equation, 241–250 equation of a loaded rigid spherical shell, 629 equation of damping oscillation, 132 equation of oscillations of the mathematical pendulum, 237 equation of the theory of diffusion boundary layer, 135 equation of transverse vibrations of a bar, 632 equation of transverse vibrations of a pointed bar, 623 equations containing arbitrary functions, 1, 116, 124, 212, 420, 518, 599, 629, 638, 656, 672 equations containing arbitrary parameters, 78, 388, 584 equations containing exponential functions, 12, 82, 118, 166, 400, 471, 577, 589, 625, 652, 666 equations containing hyperbolic functions, 15, 87, 118, 173, 407, 477, 591, 625, 667 equations containing inverse trigonometric functions, 22, 195, 503, 654, 670 equations containing logarithmic functions, 16, 89, 120, 178, 410, 487, 594, 625, 669 equations containing power functions, 4, 78, 116, 130, 388, 450, 573, 584, 617, 646, 662

equations containing trigonometric functions, 18, 90, 121, 181, 412, 490, 596, 627, 654, 670 equations of the theory of chemical reactors and the combustion theory, 56 equations of the theory of nonlinear oscillations, 56, 232 error function, 138 Euler constant, 139, 146 Euler equation, 145, 468, 624, 650 Euler formulae, 683 exponential homogeneous equation, 118, 442, 615, 679

F factorial, 690 fifth Painlevé transcendent, 398 first integrals, 247 first order linear equations, 1 first order nonlinear equations, 1–128 first Painlevé transcendent, 394 fourth order linear equations, 617–633 fourth order nonlinear equations, 633–645 fourth Painlevé transcendent, 397 fundamental system of solutions, 223, 224, 633, 661

G gamma-function, 693 Gauss hypergeometric equation, 152 Gegenbauer functions, 149 general Riccati equation, 2, 3 generalized Emden—Fowler equation, 277–301

H Hermite polynomials, 138 Heun’s equation, 157 homogeneous constant-coefficient linear equation, 647 homogeneous equation, 2, 63 homogeneous equation in the extended sense, 116, 440, 441, 614, 642, 664, 678 homogeneous linear equations of the fourth order, 617 homogeneous linear equations of the n th order, 645 homogeneous linear equations of the second order, 129

homogeneous linear equations of the third order, 449 hyperbolic functions, 684 hypergeometric equation, 152 hypergeometric series, 152, 700

I incomplete elliptic integral of the second kind, 565 incomplete gamma-function, 138 inverse hyperbolic functions, 689 inverse trigonometric functions, 687

J Jacobi elliptic function, 209, 237 Jacobi equation, 53

K Kummer’s functions, 138 Kummer’s series, 694 Kummer’s transformation, 694

L Lagrange—d’Alembert equation, 125 Laguerre polynomials, 138 Lamé equation in the form of Jacobi, 209 Lamé equation in the form of Weierstrass, 209 Laplace equation, 649 Legendre equation, 150, 161 Legendre function of the first kind, 150 Legendre function of the second kind, 150 Legendre functions, 699 Legendre polynomials, 150 Legendre transformation, 101 lemniscate functions, 271, 343 Lienard equation, 235 linear equation, 1 logarithmic derivative of the gammafunction, 139, 146, 147

M Mathieu equation, 18, 182 Mathieu functions, 183 modified associated Legendre functions, 700 modified Bessel equation, 146 modified Bessel functions, 146, 696 modified Bessel functions of the first kind, 696 modified Bessel functions of the second kind, 696 modified Emden—Fowler equation, 301–313 modified Mathieu equation, 173 movable critical points, 394

N Neumann function, 694 nonhomogeneous Bessel equation, 213 nonhomogeneous constant-coefficient linear equation, 657 nonhomogeneous Euler equation, 213, 523 nonhomogeneous linear equations of the fourth order, 617 nonhomogeneous linear equations of the n th order, 645 nonhomogeneous linear equations of the second order, 129 nonhomogeneous linear equations of the third order, 449 normal form, 57, 129

P Painlevé equations, 394 Painlevé transcendents, 394 Pochhammer symbol, 691

Q quasi-homogeneous equation, 55

R Rayleigh equation, 238 Riccati equation, 2–28

S second order linear equations, 129–227 second order nonlinear equations, 227–447 second Painlevé transcendent, 394 separable equation, 1 series solutions, 226 special functions, 693 special Riccati equation, 4

T Tchebycheff polynomials, 150 third order linear equations, 449–528 third order nonlinear equation, 528–615 third Painlevé transcendent, 396 total differential equation, 118 transcendental Painlevé functions, 394 trigonometric functions, 681

V Van der Pol equation, 239 Van der Pol oscillator, 233

W Weierstrass function, 701 Whittaker’s equation, 144 Wronskian, 646 Wronskians of the Bessel functions, 695 Wronskians of the degenerate hypergeometric functions, 698 Wronskians of the Legendre functions, 701 Wronskians of the modified Bessel functions, 697

Y Yermakov’s equation, 421