CONTENTS xxi
T5. Ordinary Differential Equations 1207
T5.1. First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207
T5.2. Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212
T5.2.1. Equations Involving Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213
T5.2.2. Equations Involving Exponential and Other Functions . . . . . . . . . . . . . . . . . . . 1220
T5.2.3. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222
T5.3. Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
T5.3.1. Equations of the Form y

xx
= f (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
T5.3.2. Equations of the Form f(x, y)y

xx
= g(x, y, y

x
) . . . . . . . . . . . . . . . . . . . . . . . . . 1225
References for Chapter T5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228
T6. Systems of Ordinary Differential Equations 1229
T6.1. Linear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
T6.1.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
T6.1.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232
T6.2. Linear Systems of Three and More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237
T6.3. Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239
T6.3.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239
T6.3.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1240
T6.4. Nonlinear Systems of Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244
References for Chapter T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246
T7. First-Order Partial Differential Equations 1247

T7.2.2. Equations of the Form
∂w
∂x
+ f(x, y, w)
∂w
∂y
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 1254
T7.2.3. Equations of the Form
∂w
∂x
+ f(x, y, w)
∂w
∂y
= g(x, y, w) . . . . . . . . . . . . . . . . . . 1256
T7.3. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258
T7.3.1. Equations Quadratic in One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258
T7.3.2. Equations Quadratic in Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259
T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives . . . . . . . . . . . . . . . . . . . 1261
References for Chapter T7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265
T8. Linear Equations and Problems of Mathematical Physics 1267
T8.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267
T8.1.1. Heat Equation
∂w
∂t
= a
∂
2
w
∂x
2

2
w
∂r
2
+
1
r
∂w
∂r

. . . . . . . . . . . . . . . 1270
T8.1.5. Equation of the Form
∂w
∂t
= a

∂
2
w
∂r
2
+
1
r
∂w
∂r

+ Φ(r, t) . . . . . . . . . . . . . . . . . . . 1271
T8.1.6. Heat Equation with Central Symmetry
∂w


+ Φ(r, t) . . . . . . . . . . . . . . . . . . . 1273
T8.1.8. Equation of the Form
∂w
∂t
=
∂
2
w
∂x
2
+
1–2β
x
∂w
∂x
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1274
xxii CONTENTS
T8.1.9. Equations of the Diffusion (Thermal) Boundary Layer . . . . . . . . . . . . . . . . . . . 1276
T8.1.10. Schr
¨
odinger Equation i
∂w
∂t
=–

2
2m
∂
2

∂x
2
+ Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1279
T8.2.3. Klein–Gordon Equation
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
– bw . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280
T8.2.4. Equation of the Form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2


∂
2
w
∂r
2
+
2
r
∂w
∂r

+ Φ(r, t) . . . . . . . . . . . . . . . . . . 1283
T8.2.7. Equations of the Form
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∂
2
w
∂x
2
+ b

∂
4
w
∂x
4
= Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1295
T8.4.3. Biharmonic Equation ΔΔw = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297
T8.4.4. Nonhomogeneous Biharmonic Equation ΔΔw = Φ(x, y) . . . . . . . . . . . . . . . . 1298
References for Chapter T8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299
T9. Nonlinear Mathematical Physics Equations 1301
T9.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301
T9.1.1. Nonlinear Heat Equations of the Form
∂w
∂t
=
∂
2
w
∂x
2
+ f(w) . . . . . . . . . . . . . . . . 1301
T9.1.2. Equations of the Form
∂w
∂t
=
∂
∂x

f(w)
∂w

∂
2
w
∂y
2
= f(w) . . . . . . . . . . . . . . . . 1318
T9.3.2. Equations of the Form
∂
∂x

f(x)
∂w
∂x

+
∂
∂y

g(y)
∂w
∂y

= f(w) . . . . . . . . . . . . . . 1321
T9.3.3. Equations of the Form
∂
∂x

f(w)
∂w
∂x

u
∂x
2
+ F (u, w),
∂w
∂t
= b
∂
2
w
∂x
2
+ G(u, w) . . . . . . 1343
T10.3.2. Systems of the Form
∂u
∂t
=
a
x
n
∂
∂x

x
n
∂u
∂x

+ F(u, w),
∂w

∂u
∂x

+ F (u, w),
∂
2
w
∂t
2
=
b
x
n
∂
∂x

x
n
∂w
∂x

+ G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368
T10.3.5. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
T10.4. Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.1. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374
T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378
T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in t . 1381
References for Chapter T10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382
T11. Integral Equations 1385

D.Sc. in 1986 from the Institute for Problems in Mechanics
of the Russian (former USSR) Academy of Sciences. Since
1975, Professor Polyanin has been working at the Institute for
Problems in Mechanics of the Russian Academy of Sciences;
he is also Professor of Mathematics at Bauman Moscow State
Technical University. He is a member of the Russian National
Committee on Theoretical and Applied Mechanics and of the
Mathematics and Mechanics Expert Council of the Higher Certification Committee of the
Russian Federation.
Professor Polyanin has made important contributions to exact and approximate analytical
methods in the theory of differential equations, mathematical physics, integral equations,
engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics.
He has obtained exact solutions for several thousand ordinary differential, partial differen-
tial, and integral equations.
Professor Polyanin is an author of more than 30 books in English, Russian, German,
and Bulgarian as well as more than 120 research papers and three patents. He has
written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev,
Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 and
2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press,
1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers
and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, and
A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis,
2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential
Equation, Chapman & Hall/CRC Press, 2004.
Professor Polyanin is editor of the book series Differential and Integral Equations
and Their Applications, Chapman & Hall/CRC Press, London/Boca Raton, and Physical
and Mathematical Reference Literature, Fizmatlit, Moscow. He is also Editor-in-Chief
of the international scientific-educational Website EqWorld—The World of Mathematical
Equations (), which is visited by over 1000 users a day worldwide.
Professor Polyanin is a member of the Editorial Board of the journal Theoretical Foundations

and the international scientific-educational Website EqWorld—The World of Mathematical
Equations ().
Professor Manzhirov has made important contributions to new mathematical methods
for solving problems in the fields of integral equations and their applications, mechanics of
growing solids, contact mechanics, tribology, viscoelasticity, and creep theory. He is the au-
thor of ten books (including Contact Problems in Mechanics of Growing Solids [in Russian],
Nauka, Moscow, 1991; Handbook of Integral Equations, CRC Press, Boca Raton, 1998;
Handbuch der Integralgleichungen: Exacte L
¨
osungen, Spektrum Akad. Verlag, Heidelberg,
1999; Contact Problems in the Theory of Creep [in Russian], National Academy of Sciences
of Armenia, Erevan, 1999), more than 70 research papers, and two patents.
Professor Manzhirov is a winner of the First Competition of the Science Support
Foundation 2001, Moscow.
Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia
Home page: />PREFACE
This book can be viewed as a reasonably comprehensive compendium of mathematical
definitions, formulas, and theorems intended for researchers, university teachers, engineers,
and students of various backgrounds in mathematics. The absence of proofs and a concise
presentation has permitted combining a substantial amount of reference material in a single
volume.
When selecting the material, the authors have given a pronounced preference to practical
aspects, namely, to formulas, methods, equations, and solutions that are most frequently
used in scientific and engineering applications. Hence some abstract concepts and their
corollaries are not contained in this book.
• This book contains chapters on arithmetics, elementary geometry, analytic geometry,
algebra, differential and integral calculus, differential geometry, elementary and special
functions, functions of one complex variable, calculus of variations, probability theory,
mathematical statistics, etc. Special attention is paid to formulas (exact, asymptotical, and
approximate), functions, methods, equations, solutions, and transformations that are of

arately in each section, while formulas (equations) and examples are numbered separately
in each subsection. When citing a formula, we use notation like (3.1.2.5), which means
xxvii