Handbook of exact solutions for ordinary differential equations:
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Chapman & Hall/CRC
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| Edition: | 2. ed. |
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| Physical Description: | XXVI, 787 S. Ill. |
| ISBN: | 1584882972 |
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| 245 | 1 | 0 | |a Handbook of exact solutions for ordinary differential equations |c Andrei D. Polyanin ; Valentin F. Zaitsev |
| 246 | 1 | 3 | |a Exact solutions for ordinary differential equations |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall/CRC |c 2003 | |
| 300 | |a XXVI, 787 S. |b Ill. | ||
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| 650 | 7 | |a Gewone differentiaalvergelijkingen |2 gtt | |
| 650 | 7 | |a Oplossingen (wiskunde) |2 gtt | |
| 650 | 4 | |a Équations différentielles - Solutions numériques | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
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| adam_text | Titel: Handbook of exact solutions for ordinary differential equations
Autor: Poljanin, Andrej D
Jahr: 2003
HANDBOOK OF
EXACT
SOLUTIONS
for ORDINARY
DIFFERENTIAL
EQUATIONS
SECOND EDITION
Andrei D. Polyanin
Valentin F. Zaitsev
CHAPMAN HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
CONTENTS
Authors ~. xxi
Foreword xxiii
Notations and Some Remarks xxv
Introduction Some Definitions, Formulas, Methods, and Transformations 1
0.1. First-Order Differential Equations 1
0.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems . 1
0.1.1-1. Equations solved for the derivative. General solution 1
0.1.1-2. The Cauchy problem. The uniqueness and existence theorems 1
0.1.1-3. Equations not solved for the derivative. The existence theorem 2
0.1.1-4. Singular solutions 2
0.1.1-5. Point transformations 2
0.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration 3
0.1.2-1. Equations with separated or separable variables 3
0.1.2-2. Equation of the form y x = f(ax + by) 3
0.1.2-3. Homogeneous equations and equations reducible to them 3
0.1.2-4. Generalized homogeneous equations and equations reducible to them . 4
0.1.2-5. Linear equation 4
0.1.2-6. Bernoulli equation 4
0.1.2-7. Equation of the form xy x =y + f(x)g(y/x) 5
0.1.2-8. Darboux equation 5
0.1.3. Exact Differential Equations. Integrating Factor 5
0.1.3-1. Exact differential equations 5
0.1.3-2. Integrating factor 5
0.1.4. Riccati Equation 6
0.1.4-1. General Riccati equation. Simplest integrable cases 6
0.1.4-2. Polynomial solutions of the Riccati equation 7
0.1.4-3. Use of particular solutions to construct the general solution 7
0.1.4-4. Some transformations 8
0.1.4-5. Reduction of the Riccati equation to a second-order linear equation .. 8
0.1.4-6. Reduction of the Riccati equation to the canonical form 8
0.1.5. Abel Equations of the First Kind 9
0.1.5-1. General form of Abel equations of the first kind. Simplest integrable
cases 9
0.1.5-2. Reduction to the canonical form. Reduction to an Abel equation of the
second kind 9
0.1.6. Abel Equations of the Second Kind 10
0.1.6-1. General form of Abel equations of the second kind. Simplest integrable
cases 10
0.1.6-2. Reduction to the canonical form. Reduction to an Abel equation of the
firstkind 11
0.1.6-3. Use of particular solutions to construct self-transformations 11
0.1.6-4. Use of particular solutions to construct the general solution 12
0.1.7. Equations Not Solved for the Derivative 14
0.1.7-1. The method of integration by differentiation. 14
0.1.7-2. Equations of the form y = f(y x) 14
vi CONTENTS
0.1.7-3. Equations of the form x = f{y x) 14
0.1.7-4. Clairaut s equation y = xy x + f(y x) 15
0.1.7-5. Lagrange s equation y = xf(y x) + g(y x) 15
0.1.8. Contact Transformations 15
0.1.8-1. General form of contact transformations 15
0.1.8-2. A method for the construction of contact transformations 16
0.1.8-3. Examples of contact transformations linear in the derivative 16
0.1.8-4. Examples of contact transformations nonlinear in the derivative 17
0.1.9. Approximate Analytic Methods for Solution of Equations 18
0.1.9-1. The method of successive approximations (Picard method) 18
0.1.9-2. The method of Taylor series expansion in the independent variable ... 18
0.1.9-3. The method of regular expansion in the small parameter 19
0.1.10. Numerical Integration of Differential Equations 20
0.1.10-1. The method of Euler polygonal lines 20
0.1.10-2. Single-step methods of the second-order approximation 20
0.1.10-3. Runge-Kutta method of the fourth-order approximation 20
0.2. Second-Order Linear Differential Equations 21
0.2.1. Formulas for the General Solution. Some Transformations 21
0.2.1-1. Homogeneous linear equations. Various representations of the general
solution 21
0.2.1-2. Wronskian determinant and Liouville s formula 21
0.2.1-3. Reduction to the canonical form 21
0.2.1-4. Reduction to the Riccati equation 22
0.2.1-5. Nonhomogeneous linear equations. The existence theorem 22
0.2.1-6. Nonhomogeneous linear equations. Various representations of the
general solution 22
0.2.1-7. Reduction to a constant coefficient equation (a special case) 22
0.2.1-8. Kummer-Liouville transformation 23
0.2.2. Representation of Solutions as a Series in the Independent Variable 23
0.2.2-1. Equation coefficients are representable in the ordinary power series
form 23
0.2.2-2. Equation coefficients have poles at some point 23
0.2.3. Asymptotic Solutions 24
0.2.3-1. Equations not containing y x. Leading asymptotic terms 24
0.2.3-2. Equations not containing y x. Two-term asymptotic expansions 25
0.2.3-3. Equations of special form not containing y x 25
0.2.3-4. Equations not containing y x. Equation coefficients are dependent on e 26
0.2.3-5. Equations containing y x 27
0.2.3-6. Equations of the general form 27
0.2.4. Boundary Value Problems 27
0.2.4-1. The first, second, third, and mixed boundary value problems 27
0.2.4-2. Simplification of boundary conditions. Reduction of equation to the
self-adjoint form 28
0.2.4-3. The Green s function. Boundary value problems for nonhomogeneous
equations 28
0.2.4-4. Representation of the Green s function in terms of particular solutions 29
0.2.5. Eigenvalue Problems 29
0.2.5-1. The Sturm-Liouville problem 29
0.2.5-2. General properties of the Sturm-Liouville problem (1), (2) 29
0.2.5-3. Problems with boundary conditions of the first kind 30
0.2.5-4. Problems with boundary conditions of the second kind 32
CONTENTS vii
0.2.5-5. Problems with boundary conditions of the third kind 33
0.2.5-6. Problems with mixed boundary conditions 33
0.3. Second-Order Nonlinear Differential Equations 33
0.3.1. Form of the General Solution. Cauchy Problem 33
0.3.1-1. Equations solved for the derivative. General solution 33
0.3.1-2. Cauchy problem. The existence and uniqueness theorem 34
0.3.2. Equations Admitting Reduction of Order 34
0.3.2-1. Equations not containing y explicitly 34
0.3.2-2. Equations not containing x explicitly (autonomous equations) 34
0.3.2-3. Equations of the form F(ax + by, y x, y x x) = 0 34
0.3.2-4. Equations of the form F(x,xy x-y,y x x) = 0 34
0.3.2-5. Homogeneous equations 35
0.3.2-6. Generalized homogeneous equations 35
0.3.2-7. Equations invariant under scaling--translation transformations 35
0.3.2-8. Exact second-order equations 36
0.3.2-9. Reduction of quasilinear equations to the normal form 37
0.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable
or Small Parameter 37
0.3.3-1. Method of expansion in powers of the independent variable 37
0.3.3-2. Method of regular (direct) expansion in powers of the small parameter 38
0.3.3-3. Pade approximants 39
0.3.4. Perturbation Methods of Mechanics and Physics 40
0.3.4-1. Preliminary remarks. A summary table of basic methods 40
0.3.4-2. The method of scaled parameters (Lindstedt-Poincare method) 40
0.3.4-3. Averaging method (Van der Pol-Krylov-Bogolyubov scheme) 42
0.3.4-4. Method of two-scale expansions (Cole-Kevorkian scheme) 43
0.3.4-5. Method of matched asymptotic expansions 44
0.3.5. Galerkin Method and Its Modifications (Projection Methods) 46
0.3.5-1. General form of an approximate solution 46
0.3.5-2. Galerkin method 47
0.3.5-3. The Bubnov-Galerkin method, the moment method, and the least
squares method 47
0.3.5-4. Collocation method 48
0.3.5-5. The method of partitioning the domain 48
0.3.5-6. The least squared error method 48
0.3.6. Iteration and Numerical Methods 49
0.3.6-1. The method of successive approximations (Cauchy problem) 49
0.3.6-2. The Runge-Kutta method (Cauchy problem) 49
0.3.6-3. Shooting method (boundary value problems) 49
0.3.6-4. Method of accelerated convergence in eigenvalue problems 50
0.4. Linear Equations of Arbitrary Order 51
0.4.1. Linear Equations with Constant Coefficients 51
0.4.1-1. Homogeneous linear equations 51
0.4.1-2. Nonhomogeneous linear equations 52
0.4.2. Linear Equations with Variable Coefficients 52
0.4.2-1. Homogeneous linear equations. Structure of the general solution . . . . 52
0.4.2-2. Utilization of particular solutions for reducing the order of die original
equation 53
0.4.2-3. Wronskian determinant and Liouville formula 54
viii CONTENTS
0.4.2-4. Nonhomogeneous linear equations. Construction of the general
solution 54
0.4.3. Asymptotic Solutions of Linear Equations 54
0.4.3-1. Fourth-order linear equations 54
0.4.3-2. Higher-order linear equations 55
0.5. Nonlinear Equations of Arbitrary Order 56
0.5.1. Structure of the General Solution. Cauchy Problem 56
0.5.1-1. Equations solved for the highest derivative. General solution 56
0.5.1-2. The Cauchy problem. The existence and uniqueness theorem 56
0.5.2. Equations Admitting Reduction of Order 56
0.5.2-1. Equations not containing y,y x,..., yx® explicitly 56
0.5.2-2. Equations not containing x explicitly (autonomous equations) 56
0.5.2-3. Equations of the form F{ax + by,y x,...,yx
n
)
) =0 57
0.5.2-4. Equations of the form F(x, xy x - y, y xx,..., y^) = 0 and its
generalizations 57
0.5.2-5. Homogeneous equations 57
0.5.2-6. Generalized homogeneous equations 57
0.5.2-7. Equations of the form F(eXx
yn
, y Jy, y ^/y y^/y) =0 58
0.5.2-8. Equations of the form F(xn
eXy
, xy x, xz
y x x xn
yx
n)
) =0 58
0.5.2-9. Other equations 58
0.5.3. A Method for Construction of Solvable Equations of General Form 59
0.5.3-1. Description of the method , 59
0.5.3-2. Examples 59
0.6. Lie Group and Discrete-Group Methods 60
0.6.1. Lie Group Method. Point Transformations 60
0.6.1-1. Local one-parameter Lie group of transformations. Invariance
condition 60
0.6.1-2. Group analysis of second-order equations. Structure of an admissible
operator 62
0.6.1-3. Utilization of local groups for reducing the order of equations and their
integration 64
0.6.2. Contact Transformations. Backlund Transformations. Formal Operators.
Factorization Principle 65
0.6.2-1. Contact transformations 65
0.6.2-2. Backlund transformations. Formal operators and nonlocal variables .. 66
0.6.2-3. Factorization principle 68
0.6.3. First Integrals (Conservation Laws) 71
0.6.4. Discrete-Group Method. Point Transformations 73
0.6.5. Discrete-Group Method. The Method of RF-Pairs 75
1. First-Order Differential Equations 81
1.1. Simplest Equations with Arbitrary Functions Integrable in Closed Form 81
1.1.1. Equations of the Form y x = f(x) 81
1.1.2. Equations of the Form y x = f(y) 81
1.1.3. Separable Equations y x = f(x)g(y) 81
1.1.4. Linear Equation g(x)y x = fi(x)y + fo(x) 81
1.1.5. Bernoulli Equation g(x)y x = fi(x)y + fn(x)yn
81
1.1.6. Homogeneous Equation y x = f(y/x) 82
CONTENTS ix
1.2. Riccati Equation g(x)y x = f2(x)y2
+ fi(x)y + fo(x) 82
1.2.1. Preliminary Remarks 82
1.2.2. Equations Containing Power Functions ; 82
1.2.2-1. Equations of the form g(x)y x = f2(x)y2
+ fo(x) 82
1.2.2-2. Other equations 84
1.2.3. Equations Containing Exponential Functions 89
1.2.3-1. Equations with exponential functions 89
1.2.3-2. Equations with power and exponential functions 90
1.2.4. Equations Containing Hyperbolic Functions 92
1.2.4-1. Equations widi hyperbolic sine and cosine 92
1.2.4-2. Equations widi hyperbolic tangent and cotangent 93
1.2.5. Equations Containing Logarithmic Functions 94
1.2.5-1. Equations of the form g(x)y x = fi(x)y2
+ fo(x) 94
1.2.5-2. Equations of the form g(x)y x = fi(x)y2
+ f (x)y x + fo(x) 94
1.2.6. Equations Containing Trigonometric Functions 95
1.2.6-1. Equations with sine 95
1.2.6-2. Equations with cosine 96
1.2.6-3. Equations with tangent 97
1.2.6-4. Equations with cotangent 98
1.2.6-5. Equations containing combinations of trigonometric functions 99
1.2.7. Equations Containing Inverse Trigonometric Functions 100
1.2.7-1. Equations containing arcsine 100
1.2.7-2. Equations containing arccosine 100
1.2.7-3. Equations containing arctangent 101
1.2.7-4. Equations containing arccotangent 101
1.2.8. Equations with Arbitrary Functions 102
1.2.8-1. Equations containing arbitrary functions (but not containing their
derivatives) 102
1.2.8-2. Equations containing arbitrary functions and their derivatives 104
1.2.9. Some Transformations 105
1.3. Abel Equations of the Second Kind 107
1.3.1. Equations of the Form yy x-y = f{x) 107
1.3.1-1. Preliminary remarks. Classification tables 107
1.3.1-2. Solvable equations and their solutions 109
1.3.2. Equations of the Form yy x = f(x)y + 1 120
1.3.3. Equations of the Form yy x = f (x)y + fo(x) 121
1.3.3-1. Preliminary remarks 121
1.3.3-2. Solvable equations and their solutions 121
1.3.4. Equations of the Form [g (x)y + go(x)]y x = f2(x)y2
+ fi(x)y + fo(x) 132
1.3.4-1. Preliminary remarks 132
1.3.4-2. Solvable equations and their solutions 132
1.3.5. Some Types of First- and Second-Order Equations Reducible to Abel Equations
of the Second Kind 136
1.3.5-1. Quasi-homogeneous equations 136
1.3.5-2. Equations of the theory of chemical reactors and the combustion theory 136
1.3.5-3. Equations of the theory of nonlinear oscillations 136
1.3.5-4. Second-order homogeneous equations of various types 137
1.3.5-5. Second-order equations invariant under some transformations 137
CONTENTS
1.4. Equations Containing Polynomial Functions of y 138
1.4.1. Abel Equations of the First Kind y x = f3(x)y3
+ h(x)y2
+ fi(x)y + fo(x) 138
1.4.1-1. Preliminary remarks 138
1.4.1-2. Solvable equations and their solutions 138
1.4.2. EquationsoftheForm(A22y2
+An%y+Aux2
+Ao)yx=B22y2
+Bi2xy+Bnx2
+Bo 142
1.4.2-1. Preliminary remarks. Some transformations 142
1.4.2-2. Solvable equations and their solutions 143
1.4.3. Equations of the Form (A222/2
+ Mixy + A x2
+ A2y + Axx)y x =
B22I)1
+ Bnxy + Bnx2
+ B2y + Bxx 144
1.4.3-1. Preliminary remarks 144
1.4.3-2. Solvable equations and their solutions 145
1.4.4. Equations of the Form (A222/2
+ A 2xy + A x2
+ A2y + A x + Ao)y x =
B22y2
+ Bnxy + Bnx2
+ B2y + Bxx + B0 151
1.4.4-1. Preliminary remarks. Some transformations 151
1.4.4-2. Solvable equations and their solutions 152
1.4.5. Equations of the Form (A3y3
+ A2xy2
+ A xz
y + AQX3
+ a y + aox)y x =
B3y3
+ B2xy2
+ Bix2
y + B0x3
+bxy + box 155
1.5. Equations of the Form f(x, y)y x = g(x, y) Containing Arbitrary Parameters 159
1.5.1. Equations Containing Power Functions 159
1.5.1-1. Equations of the form y x = f(x,y) 159
1.5.1-2. Other equations 160
1.5.2. Equations Containing Exponential Functions 162
1.5.2-1. Equations with exponential functions 162
1.5.2-2. Equations with power and exponential functions 163
1.5.3. Equations Containing Hyperbolic Functions 166
1.5.4. Equations Containing Logarithmic Functions 168
1.5.5. Equations Containing Trigonometric Functions 169
1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic,
and Trigonometric Functions 171
1.6. Equations of the Form F(x,y,y x) = 0 Containing Arbitrary Parameters 173
1.6.1. Equations of the Second Degree in y x 173
1.6.1-L Equations of the form f(x,y)(y x)2
= g(x,y) 173
1.6.1-2. Equations of the form f(x, y)(y x)2
= g(x, y)y x + h(x, y) 175
1.6.2. Equations of the Third Degree in y x 180
1.6.2-1. Equations of the form f(x, y)(y x)3
= g(x, y)y x + h(x, y) 180
1.6.2-2. Equations of the form f(x, y)(y x)3
= g(x, y)(y x)2
+ h(x, y)y x + r(x, y) 180
1.6.3. Equations of the Form (y x)k
= f(y) + g(x) 182
1.6.3-1. Some transformations 182
1.6.3-2. Classification tables and exact solutions 183
1.6.4. Other Equations 190
1.6.4-1. Equations containing algebraic and power functions with respect to y x 190
1.6.4-2. Equations containing exponential, logarithmic, and other functions with
respect to y x 193
1.7. Equations of the Form f(x, y)y x = g(x, y) Containing Arbitrary Functions 195
1.7.1. Equations Containing Power Functions 195
1.7.2. Equations Containing Exponential and Hyperbolic Functions 197
1.7.3. Equations Containing Logarithmic Functions 199
1.7.4. Equations Containing Trigonometric Functions 200
1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and
Trigonometric Functions 201
CONTENTS xi
1.8. Equations of the Form F(x, y, y x) = 0 Containing Arbitrary Functions 203
1.8.1. Some Equations 203
1.8.1-1. Arguments of arbitrary functions depend on x and y 203
1.8.1-2. Argument of arbitrary functions is y x 204
1.8.1-3. Arguments of arbitrary functions are linear witfi respect to y x 205
1.8.1-4. Arguments of arbitrary functions are nonlinear with respect to y x . . . . 209
1.8.2. Some Transformations 212
2. Second-Order Differential Equations 213
2.1. Linear Equations 213
2.1.1. Representation of the General Solution Through a Particular Solution 213
2.1.2. Equations Containing Power Functions 213
2.1.2-1. Equations of the form y ^x + f(x)y =0 213
2.1.2-2. Equations of the form y lx + j(x)y x + g(x)y = 0 215
2.1.2-3. Equations of the form (ax + b)yxx + f{x)y x + g(x)y = 0 219
2.1.2-4. Equations of the form x2
yxx + f(x)y x + g{x)y = 0 225
2.1.2-5. Equations of the form (ax2
+bx + c)yxx + f(x)y x + g{x)y = 0 230
2.1.2-6. Equations of the form (a,3X3
+ a2X2
+ aix + ao)yxx+f(x)yx+g(x)y = Q 237
2.1.2-7. Equations of the form (a4x4
+ - · · + axx + ao)yxx + f(x)y x + g(x)y = 0 240
2.1.2-8. Other equations 244
2.1.3. Equations Containing Exponential Functions 246
2.1.3-1. Equations with exponential functions 246
2.1.3-2. Equations with power and exponential functions 250
2.1.4. Equations Containing Hyperbolic Functions 252
2.1.4-1. Equations with hyperbolic sine 252
2.1.4-2. Equations with hyperbolic cosine 253
2.1.4-3. Equations with hyperbolic tangent 254
2.1.4-4. Equations with hyperbolic cotangent 255
2.1.4-5. Equations containing combinations of hyperbolic functions 256
2.1.5. Equations Containing Logariuimic Functions 257
2.1.5-1. Equations of the form f(x)yxx + g(x)y = 0 257
2.1.5-2. Equations of the form f(x)yxx + g(x)y x + h(x)y = 0 258
2.1.6. Equations Containing Trigonometric Functions 260
2.1.6-1. Equations with sine 260
2.1.6-2. Equations with cosine 262
2.1.6-3. Equations with tangent 265
2.1.6-4. Equations with cotangent 267
2.1.6-5. Equations containing combinations of trigonometric functions 269
2.1.7. Equations Containing Inverse Trigonometric Functions 271
2.1.7-1. Equations with arcsine 271
2.1.7-2. Equations with arccosine 273
2.1.7-3. Equations with arctangent 274
2.1.7-4. Equations with arccotangent 276
2.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric,
and Other Functions 277
2.1.9. Equations with Arbitrary Functions 285
2.1.9-1. Equations containing arbitrary functions (but not containing their
derivatives) 285
2.1.9-2. Equations containing arbitrary functions and their derivatives 289
2.1.10. Some Transformations 292
xii CONTENTS
2.2. Autonomous Equations y xx = F{y, y x) 295
2.2.1. Equations of the Form y%x -y x = f(y) 295
2.2.2. Equations of the Form yxx+f(y)y x + y = 0 299
2.2.2-1. Preliminary remarks 299
2.2.2-2. Solvable equations and their solutions 299
2.2.3. LienardEquations y x x + f(y)y x+g(y) = 0 301
2.2.3-1. Preliminary remarks 301
2.2.3-2. Solvable equations and their solutions 302
2.2.4. Rayleigh Equations yxx + f(y x) + g(y) = 0 304
2.2.4-1. Preliminary remarks. Some transformations 304
2.2.4-2. Solvable equations and their solutions 305
2.3. Emden-Fowler Equation y x x = Axn
ym
306
2.3.1. Exact Solutions 306
2.3.1-1. Preliminary remarks. Classification table 306
2.3.1-2. Solvable equations and their solutions 307
2.3.2. First Integrals (Conservation Laws) 312
2.3.2-1. First integrals withfc= 2 312
2.3.2-2. First integrals with k = 3 312
2.3.2-3. First integrals with k = 4 313
2.3.2-4. First integrals with k = 5 314
2.3.3. Some Formulas and Transformations 314
2.4. Equations of the Form y x x = Aixni
ymi
+ A2xni
ym2
314
2.4.1. Classification Table 314
2.4.2. Exact Solutions 318
2.5. Generalized Emden-Fowler Equation yxx = Axn
ym
(y x)1
336
2.5.1. Classification Table 336
2.5.2. Exact Solutions 339
2.5.3. Some Formulas and Transformations 354
2.5.3-1. A particular solution 354
2.5.3-2. Discrete transformations of die generalized Emden-Fowler equation . 355
2.5.3-3. Reduction of the generalized Emden-Fowler equation to an Abel
equation 355
2.6. Equations of the Form y x x = Aixnr
ymi
(y x)h
+ A2xn2
ymi
(y x)h
356
2.6.1. Modified Emden-Fowler Equation y x x = Aix^y^ + A2xn
ym
356
2.6.1-1. Preliminary remarks. Classification table 356
2.6.1-2. Solvable equations and their solutions 358
2.6.2. Equations of the Form y x x = (Aixni
ym
+A2Xn2
ymi
)(y x)1
365
2.6.2-1. Classification table 365
2.6.2-2. Solvable equations and their solutions 370
2.6.3. Equations of the Form yx x = aAxn
ym
(y J + Axn
-l
ym+1
(y x)1
-1
393
2.6.3-1. Classification table 393
2.6.3-2. Solvable equations and their solutions 396
2.6.4. Other Equations (lx*l2) 406
2.6.4-1. Classification table 406
2.6.4-2. Solvable equations and their solutions 407
2.7. Equations of the Form y x x - f(x)g(y)h(y x) 411
2.7.1. Equations of the Form y x x = f(x)g(y) 412
2.7.2. Equations Containing Power Functions (h £ const) 414
2.7.3. Equations Containing Exponential Functions (h £ const) 418
CONTENTS xiii
2.7.3-1. Preliminary remarks 418
2.7.3-2. Solvable equations and their solutions 419
2.7.4. Equations Containing Hyperbolic Functions (h £ const) 421
2.7.5. Equations Containing Trigonometric Functions (h £ const) 423
2.7.6. Some Transformations 424
2.8. Some Nonlinear Equations with Arbitrary Parameters 425
2.8.1. Equations Containing Power Functions 425
2.8.1-1. Equations of the form f(x, y)yxx + g(x, y) = 0 425
2.8.1-2. Equations of the form f(x,y)y x
l
x+g(x,y)y x + h(x,y) = O 427
2.8.1-3. Equations of the form f(x,y)yx x+g(x,y)(yx)2
+ h(x,y)y x+r(x,y) = 0 428
2.8.1-4. Other equations 430
2.8.2. Painleve Transcendents 432
2.8.2-1. Preliminary remarks. Singular points of solutions 432
2.8.2-2. First Painleve transcendent 432
2.8.2-3. Second Painleve transcendent 433
2.8.2-4. Third Painleve transcendent 434
2.8.2-5. Fourth Painleve transcendent 435
2.8.2-6. Fifth Painleve transcendent 436
2.8.2-7. Sixth Painleve transcendent 437
2.8.3. Equations Containing Exponential Functions 438
2.8.3-1. Equations of the form f(x, y)y x x + g(x, y) = 0 438
2.8.3-2. Equations of the form f(x, y)y x x + g(x, y)y x + h(x, y) = 0 438
2.8.3-3. Equations of the form f(x,y)y x
l
x+g(x,y)(yl
x)2
+ h(x,y)yx+r(x,y) = 0 441
2.8.3-4. Other equations 443
2.8.4. Equations Containing Hyperbolic Functions 445
2.8.4-1. Equations with hyperbolic sine 445
2.8.4-2. Equations with hyperbolic cosine 447
2.8.4-3. Equations with hyperbolic tangent 448
2.8.4-4. Equations widi hyperbolic cotangent 449
2.8.4-5. Equations containing combinations of hyperbolic functions 450
2.8.5. Equations Containing Logarithmic Functions 450
2.8.5-1. Equations of me form f(x, y)yxx + g(x, y)y x + h(x, y) = 0 450
2.8.5-2. Other equations 451
2.8.6. Equations Containing Trigonometric Functions 452
2.8.6-1. Equations with sine 452
2.8.6-2. Equations with cosine 454
2.8.6-3. Equations with tangent 456
2.8.6-4. Equations with cotangent 457
2.8.6-5. Equations containing combinations of trigonometric functions 458
2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic,
and Trigonometric Functions 459
2.9. Equations Containing Arbitrary Functions 461
2.9.1. Equations of the Form F(x, y)y xx + G(x,.y) = 0 461
2.9.1-1. Arguments of arbitrary functions are algebraic and power functions of
x and y 461
2.9.1-2. Arguments of the arbitrary functions are other functions 465
2.9.2. Equations of the Form F(x, y)yxx + G(x, y)y x + H(x, y) = 0 467
2.9.2-1. Argument of the arbitrary functions is a; 467
2.9.2-2. Argument of the arbitrary functions is y 468
2.9.2-3. Other arguments of the arbitrary functions 469
xiv CONTENTS
M
2.9.3. Equations of the Form F(x, y)y x x + £ Gm(x, y)(y x)m
= 0 ( M = 2, 3, 4) . . . . 471
m=0
2.9.3-1. Argument of the arbitrary functions is x 471
2.9.3-2. Argument of the arbitrary functions is y 473
2.9.3-3. Other arguments of arbitrary functions 474
2.9.4. Equations of the Form F(x, y, y x)yxx + G(x, y,y x) = 0 475
2.9.4-1. Arguments of the arbitrary functions depend on x or y 475
2.9.4-2. Arguments of die arbitrary functions depend on x and y 476
2.9.4-3. Arguments of die arbitrary functions depend on x, y, and y x 479
2.9.5. Equations Not Solved for Second Derivative 484
2.9.6. Equations of General Form 486
2.9.6-1. Equations containing arbitrary functions of two variables 486
2.9.6-2. Equations containing arbitrary functions of three variables 491
2.9.7. Some Transformations 492
3. Third-Order Differential Equations 495
3.1. Linear Equations 495
3.1.1. Preliminary Remarks 495
3.1.2. Equations Containing Power Functions 496
3.1.2-1. Equations of the form /3(x)2/^x + fo(x)y = g(x) 496
3.1.2-2. Equations of the form f3(x)y x xx + fi(x)y x + fo(x)y = g(x) 497
3.1.2-3. Equations of the form f3(x)y x xx + /2 W * + /i (x)y x + fo(x)y = g(x) 503
3.1.3. Equations Containing Exponential Functions 512
3.1.3-1. Equations with exponential functions 512
3.1.3-2. Equations with power and exponential functions 515
3.1.4. Equations Containing Hyperbolic Functions 516
3.1.4-1. Equations widi hyperbolic sine 516
3.1.4-2. Equations with hyperbolic cosine 518
3.1.4-3. Equations with hyperbolic sine and cosine 520
3.1.4-4. Equations with hyperbolic tangent 520
3.1.4-5. Equations with hyperbolic cotangent 523
3.1.5. Equations Containing Logarithmic Functions 525
3.1.5-1. Equations with logarithmic functions 525
3.1.5-2. Equations with power and logarithmic functions 526
3.1.6. Equations Containing Trigonometric Functions 528
3.1.6-1. Equations widi sine 528
3.1.6-2. Equations widi cosine 531
3.1.6-3. Equations with sine and cosine 533
3.1.6-4. Equations with tangent 534
3.1.6-5. Equations widi cotangent 537
3.1.7. Equations Containing Inverse Trigonometric Functions 539
3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric,
and Otiier Functions 544
3.1.9. Equations Containing Arbitrary Functions 550
3.1.9-1. Equations of the form h(x)y x xx + fi(x)y x + fo(x)y = g(x) 550
3.1.9-2. Equations of the form / 3 ( a ; ) C + h(x)y x x + fi(x)y x + fo(x)y = g(x) 553
3.2. Equations of the Form yx xx = Axa
y0
(.yxy(yx x)s
559
3.2.1. Classification Table 559
3.2.2. Equations of the Form y J. ^ = Ay0
566
CONTENTS XV
3.2.3. Equations of the Form y^xx = Axa
y0
567
3.2.4. Equations with hi + 6 * 0 568
3.2.5. Some Transformations 592
3.3. Equations of the Form y x xx = Hy)g(y xMy x x) 592
3.3.1. Equations Containing Power Functions 592
3.3.2. Equations Containing Exponential Functions 595
3.3.3. Other Equations 599
3.4. Nonlinear Equations with Arbitrary Parameters 601
3.4.1. Equations Containing Power Functions 601
3.4.1-1. Equations of die form f(x,y)y xxx = g(x,y) 601
3.4.1-2. Equations of the form y x xx = f(x, y,y x) 602
3.4.1-3. Equations of the form f(x, y, y x)y x xx + g(x, y, yx)y x x + h(x, y, y x) = 0 603
3.4.1-4. Otiier equations 607
3.4.2. Equations Containing Exponential Functions 608
3.4.2-1. Equations of the form y x xx = fix, y,y x) 608
3.4.2-2. Other equations 609
3.4.3. Equations Containing Hyperbolic Functions 611
3.4.3-1. Equations with hyperbolic sine 611
3.4.3-2. Equations with hyperbolic cosine 612
3.4.3-3. Equations with hyperbolic tangent 613
3.4.3-4. Equations with hyperbolic cotangent 615
3.4.4. Equations Containing Logarithmic Functions 616
3.4.4-1. Equations of the form y x xx = f(x, y, y x) 616
3.4.4-2. Other equations 617
3.4.5. Equations Containing Trigonometric Functions 618
3.4.5-1. Equations widi sine 618
3.4.5-2. Equations with cosine 619
3.4.5-3. Equations with tangent 620
3.4.5-4. Equations widi cotangent 621
3.5. Nonlinear Equations Containing Arbitrary Functions 622
3.5.1. Equations of the Form F(x,y)y x xx + G(x,y) = 0 622
3.5.1-1. Arguments of the arbitrary functions are x or y 622
3.5.1-2. Arguments of die arbitrary functions depend on x and y 623
3.5.2. Equations of the Form F(x, y, yx)yxxx + G(x, y,y x) = 0 624
3.5.2-1. Arguments of die arbitrary functions depend on x and y 624
3.5.2-2. Arguments of the arbitrary functions depend on x, y, and y x 625
3.5.3. Equations of the Form F(x, y, y x)yx xx + G(x, y, y x)y x x + H(x, y,y x) = 0 629
3.5.3-1. The arbitrary functions depend on x or y 629
3.5.3-2. Arguments of arbitrary functions depend on x and y 632
3.5.3-3. Arguments of arbitrary functions depend on x, y, and y x 633
3.5.4. Equations of the Form F(x, y, y Jy ^x + £ Ga(x, y, y Mx)a
=0 634
a
3.5.4-1. Arbitrary functions depend on x or y 634
3.5.4-2. Arguments of arbitrary functions depend on x, y, and y x 635
3.5.5. Other Equations 636
3.5.5-1. Equations of the form F(x, y, y x, y L)y x xx+G(x,y,yl
x,y x x) = Q . . . . 636
3.5.5-2. Equations of the form F(x, y, y x, yxx,y x xx) = 0 638
xvi CONTENTS
4. Fourth-Order Differential Equations 641
4.1. Linear Equations 641
4.1.1. Preliminary Remarks 641
4.1.2. Equations Containing Power Functions 641
4.1.2-1. Equations of the form f4(x)y x xxx + fo(x)y = g(x) 641
4.1.2-2. Equations of the form f4ix)y x x xx + /i (x)y x + fo(x)y = g(x) 642
4.1.2-3. Equations of the form f4(x)y x x
l
xx + f2(x)y x x + fl(x)y x + Mx)y = g(x) 644
4.1.2-4. Other equations 645
4.1.3. Equations Containing Exponential and Hyperbolic Functions 648
4.1.3-1. Equations with exponential functions 648
4.1.3-2. Equations widi hyperbolic functions 649
4.1.4. Equations Containing Logarithmic Functions 651
4.1.5. Equations Containing Trigonometric Functions 652
4.1.5-1. Equations with sine and cosine 652
4.1.5-2. Equations with tangent and cotangent 654
4.1.6. Equations Containing Arbitrary Functions 655
4.1.6-1. Equations of the form Mx)yxxxx + fi(x)y x + fo(x)y = g(x) 655
4.1.6-2. Equations of the form f4(x)y x x xx + f1ix)yl
x x + /i (x)y x + fQ(x)y = g(x) 657
4.1.6-3. Otiier equations 657
4.2. Nonlinear Equations 659
4.2.1. Equations Containing Power Functions 659
4.2.1-1. Equations of the form y ^ = f(x,y) 659
4.2.1-2. Equations of the form y^ x x = f(x, y,y x) 660
4.2.1-3. Equations of the form y ^ = fix, y,y x,yxx) 660
4.2.1-4. Equations of the form y^xx = fix, y, y x, y ^, y £x) 663
4.2.2. Equations Containing Exponential Functions 666
4.2.2-1. Equations of the form yxxxx = fix, y) 666
4.2.2-2. Other equations 666
4.2.3. Equations Containing Hyperbolic Functions 668
4.2.3-1. Equations with hyperbolic sine 668
4.2.3-2. Equations with hyperbolic cosine 669
4.2.3-3. Equations with hyperbolic tangent 671
4.2.3-4. Equations with hyperbolic cotangent 671
4.2.4. Equations Containing Logaritiimic Functions 672
4.2.4-1. Equations of the form y ^xx = f(x,y) 672
4.2.4-2. Other equations 673
4.2.5. Equations Containing Trigonometric Functions 674
4.2.5-1. Equations with sine 674
4.2.5-2. Equations widi cosine 675
4.2.5-3. Equations widi tangent 677
4.2.5-4. Equations with cotangent 677
4.2.6. Equations Containing Arbitrary Functions 678
4.2.6-1. Equations of the form y J. ^ = fix,y) 678
4.2.6-2. Equations of the form y^ x x = fix, y,y x) 680
4.2.6-3. Equations of the form y ^xx = f(x,y,y x,y J.x) 681
4.2.6-4. Equations of the form y ^xx = f(x, y, y x, y ^, y x xx) 683
4.2.6-5. Other equations 686
CONTENTS xvii
5. Higher-Order Differential Equations 689
5.1. Linear Equations 689
5.1.1. Preliminary Remarks 689
5.1.2. Equations Containing Power Functions 689
5.1.2-1. Equations of the form fnix)yx
n)
+ fo(x)y = g(x) 689
5.1.2-2. Equations of the form fn(x)yx
n)
+ f (x)y x + fo(x)y = g(x) 692
5.1.2-3. Other equations 692
5.1.3. Equations Containing Exponential and Hyperbolic Functions 695
5.1.3-1. Equations widi exponential functions 695
5.1.3-2. Equations widi hyperbolic functions 696
5.1.4. Equations Containing Logaridimic Functions 698
5.1.5. Equations Containing Trigonometric Functions 698
5.1.5-1. Equations widi sine and cosine 698
5.1.5-2. Equations with tangent and cotangent 700
5.1.6. Equations Containing Arbitrary Functions 701
5.1.6-1. Equations of the form fnix)yx
n)
+ f ix)y x + foix)y = gix) 701
5.1.6-2. Other equations 702
5.2. Nonlinear Equations 705
5.2.1. Equations Containing Power Functions 705
5.2.1-1. Fifdi- and sixdi-order equations 705
5.2.1-2. Equations of the form yx
n)
= fix, y) 706
5.2.1-3. Equations of the form yx
n)
= fix, y, y x, yx x) 707
5.2.1-4. Other equations 708
5.2.2. Equations Containing Exponential Functions 711
5.2.2-1. Fifth- and sixth-order equations 711
5.2.2-2. Equations of the form yx
n)
= fix, y) 712
5.2.2-3. Otiier equations 712
5.2.3. Equations Containing Hyperbolic Functions 713
5.2.3-1. Equations with hyperbolic sine 713
5.2.3-2. Equations with hyperbolic cosine 714
5.2.3-3. Equations with hyperbolic tangent 715
5.2.3-4. Equations widi hyperbolic cotangent 716
5.2.4. Equations Containing Logaridimic Functions 717
5.2.4-1. Equations of the form y^ = f(x, y) 717
5.2.4-2. Other equations 718
5.2.5. Equations Containing Trigonometric Functions 718
5.2.5-1. Equations with sine 718
5.2.5-2. Equations with cosine 719
5.2.5-3. Equations with tangent 720
5.2.5-4. Equations with cotangent 721
5.2.6. Equations Containing Arbitrary Functions 722
5.2.6-1. Fifth- and sixth-order equations 722
5.2.6-2. Equations of the form yx
n)
= fix, y) 723
5.2.6-3. Equations of the form y .n)
= fix, y,y x) 726
5.2.6-4. Equations of the form 2/x
n)
= fix, y, y x, yxx) 727
5.2.6-5. Equations of the form fix,y)yx
n)
+ g(x,y,y x)yx
n
-1)
=
fc(z,2/,2/x,...,^-2
) 728
5.2.6-6. Equations of the form yx
n)
= / (x, y, y x,..., 2/£v
1)
) 730
5.2.6-7. Equations of the general form F (a;, j/,2/x,...,2/x
n)
) =0 731
xviii CONTENTS
Supplements 735
5.1. Elementary Functions and Their Properties 735
5.1.1. Trigonometric Functions 735
S.I.1-1. Simplest relations 735
S.I.1-2. Relations between trigonometric functions of single argument 735
S.I.1-3. Reduction formulas 735
S.I. 1-4. Addition and subtraction of trigonometric functions 736
S.I.1-5. Products of trigonometric functions 736
S.I.1-6. Powers of trigonometric functions 736
S.I.1-7. Addition formulas 737
S.I.1-8. Trigonometric functions of multiple arguments 737
S.I.1-9. Trigonometric functions of half argument 737
S. 1.1-10. Euler and de Moivre formulas. Relationship with hyperbolic functions 737
S.I.1-11. Differentiation formulas 737
S.I.1-12. Expansion into power series 738
5.1.2. Hyperbolic Functions 738
S.l.2-1. Definitions 738
S.l.2-2. Simplest relations 738
S. 1.2-3. Relations between hyperbolic functions of single argument (x 0) .. 738
S.l.2-4. Addition formulas 738
S. 1.2-5. Addition and subtraction of hyperbolic functions 739
S. 1.2-6. Products of hyperbolic functions 739
S.1.2-7. Powers of hyperbolic functions 739
S. 1.2-8. Hyperbolic functions of multiple arguments 739
S.I.2-9. Relationship with trigonometric functions 740
S.1.2-10. Differentiation formulas 740
S. 1.2-11. Expansion into power series 740
5.1.3. Inverse Trigonometric Functions 740
S. 1.3-1. Definitions and some properties 740
S.l.3-2. Simplest formulas 741
S. 1.3-3. Relations between inverse trigonometric functions 741
S. 1.3-4. Addition and subtraction of inverse trigonometric functions 741
S.l.3-5. Differentiation formulas 741
S.1.3-6. Expansion into power series 742
5.1.4. Inverse Hyperbolic Functions 742
S. 1.4-1. Relationships with logaridimic functions 742
S. 1.4-2. Relations between inverse hyperbolic functions 742
S. 1.4-3. Addition and subtraction of inverse hyperbolic functions 742
S.1.4-4. Differentiation formulas 742
S. 1.4-5. Expansion into power series 743
5.2. Special Functions and Their Properties 743
5.2.1. Some Symbols and Coefficients 743
S.2.1-1. Factorials 743
S.2.1-2. Binomial coefficients 743
S.2.1-3. Pochhammer symbol 744
5.2.2. Error Functions and Exponential Integral 744
S.2.2-1. Error function and complementary error function 744
S.2.2-2. Exponential integral 744
S.2.2-3. Logarithmic integral 745
CONTENTS xix
5.2.3. Gamma and Beta Functions 745
S.2.3-1. Gamma function 745
S.2.3-2. Logarithmic derivative of the gamma function 746
S.2.3-3. Beta function 747
5.2.4. Incomplete Gamma and Beta Functions 747
S.2.4-1. Incomplete gamma function 747
S.2.4-2. Incomplete beta function 747
5.2.5. Bessel Functions 748
S.2.5-1. Definitions and basic formulas 748
S.2.5-2. Bessel functions for v = ±n± , where n = 0, 1, 2 748
S.2.5-3. Bessel functions for v = ±n, where n = 0, 1, 2, 749
S.2.5-4. Wronskians and similar formulas 749
S.2.5-5. Integral representations 749
S.2.5-6. Asymptotic expansions 750
S.2.5-7. Zeros and orthogonality properties of die Bessel functions 750
S.2.5-8. Hankel functions (Bessel functions of the diird kind) 750
5.2.6. Modified Bessel Functions 751
S.2.6-1. Definitions. Basic formulas 751
S.2.6-2. Modified Bessel functions for v = ±n± , where n = 0, 1, 2, 751
S.2.6-3. Modified Bessel functions for v = n, where n = 0, 1, 2, 752
S.2.6-4. Wronskians and similar formulas 752
S.2.6-5. Integral representations 752
S.2.6-6. Asymptotic expansions as x -- oo 752
5.2.7. Degenerate Hypergeometric Functions 753
S.2.7-1. Definitions. The Kummer s series 753
S.2.7-2. Some transformations and linear relations 753
S.2.7-3. Differentiation formulas and Wronskian 753
S.2.7-4. Degenerate hypergeometric functions for n = 0, 1, 2, 754
S.2.7-5. Integral representations 754
S.2.7-6. Asymptotic expansion as x - oo 754
S.2.7-7. Whittaker functions 755
5.2.8. Hypergeometric Functions 755
S.2.8-1. Definition. The hypergeometric series 755
S.2.8-2. Basic properties 755
S.2.8-3. Integral representations 755
5.2.9. Legendre Functions and Legendre Polynomials 756
S.2.9-1. Definitions. Basic formulas 756
S.2.9-2. Trigonometric expansions 756
S.2.9-3. Some relations 757
S.2.9-4. Integral representations 757
S.2.9-5. Legendre polynomials 757
S.2.9-6. Zeros of the Legendre polynomials and die generating function 758
S.2.9-7. Associated Legendre functions 758
5.2.10. Parabolic Cylinder Functions 758
S.2.10-1. Definitions. Basic formulas 758
S.2.10-2. Integral representations 759
S.2.10-3. Asymptotic expansion as z -* oo 759
5.2.11. Orthogonal Polynomials 759
S.2.11-1. Laguerre polynomials and generalized Laguerre polynomials 759
S.2.11-2. Chebyshev polynomials 760
S.2.11-3. Hermite polynomials 761
xx CONTENTS
S.2.11-4. Gegenbauer polynomials 762
S.2.11-5. Jacobi polynomials 762
S.2.12. The Weierstrass Function 762
S.2.12-1. Definitions 762
S.2.12-2. Some properties 762
S.3. Tables of Indefinite Integrals 763
5.3.1. Integrals Containing Rational Functions 763
S.3.1-1. Integrals containing a + bx 763
S.3.1-2. Integrals containing a + x and b + x 763
S.3.1-3. Integrals containing a2
+ x2
764
S.3.1-4. Integrals containing a2
-x2
765
S.3.1-5. Integrals containing a3
+x3
765
S.3.1-6. Integrals containing a3
- x3
766
S.3.1-7. Integrals containing a4
+ a;4
766
5.3.2. Integrals Containing Irrational Functions 767
S.3.2-1. Integrals containing xx
l2
767
S.3.2-2. Integrals containing (a + bxfl2
767
S.3.2-3. Integrals containing (a:2
+ a2
)1
/2
768
S.3.2-4. Integrals containing (a;2
- a2
)1
/2
768
S.3.2-5. Integrals containing (a2
- x2
)1
/2
768
S.3.2-6. Reduction formulas 769
5.3.3. Integrals Containing Exponential Functions 769
5.3.4. Integrals Containing Hyperbolic Functions 769
S.3.4-1. Integrals containing cosha; 769
S.3.4-2. Integrals containing sinhx 770
S.3.4-3. Integrals containing tanhx or cotha; 771
5.3.5. Integrals Containing Logarithmic Functions 772
5.3.6. Integrals Containing Trigonometric Functions 773
S.3.6-1. Integrals containing cos x 773
S.3.6-2. Integrals containing sin x 774
S.3.6-3. Integrals containing sin a; and cos a; 776
S.3.6-4. Reduction formulas 776
S.3.6-5. Integrals containing tana; and cota; 776
5.3.7. Integrals Containing Inverse Trigonometric Functions 777
References 779
Index * 783
|
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| author | Poljanin, Andrej D. 1951- Zajcev, Valentin F. |
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| callnumber-first | Q - Science |
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| dewey-full | 515/.352 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.352 |
| dewey-search | 515/.352 |
| dewey-sort | 3515 3352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV014547911 |
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| indexdate | 2024-07-09T19:03:22Z |
| institution | BVB |
| isbn | 1584882972 |
| language | English |
| lccn | 2002073735 |
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| spelling | Poljanin, Andrej D. 1951- Verfasser (DE-588)128391251 aut Handbook of exact solutions for ordinary differential equations Andrei D. Polyanin ; Valentin F. Zaitsev Exact solutions for ordinary differential equations 2. ed. Boca Raton [u.a.] Chapman & Hall/CRC 2003 XXVI, 787 S. Ill. txt rdacontent n rdamedia nc rdacarrier Gewone differentiaalvergelijkingen gtt Oplossingen (wiskunde) gtt Équations différentielles - Solutions numériques Differential equations Numerical solutions Exakte Lösung (DE-588)4348289-2 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Exakte Lösung (DE-588)4348289-2 s DE-604 Zajcev, Valentin F. Verfasser (DE-588)12839126X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009892069&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Poljanin, Andrej D. 1951- Zajcev, Valentin F. Handbook of exact solutions for ordinary differential equations Gewone differentiaalvergelijkingen gtt Oplossingen (wiskunde) gtt Équations différentielles - Solutions numériques Differential equations Numerical solutions Exakte Lösung (DE-588)4348289-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4348289-2 (DE-588)4020929-5 (DE-588)4155008-0 |
| title | Handbook of exact solutions for ordinary differential equations |
| title_alt | Exact solutions for ordinary differential equations |
| title_auth | Handbook of exact solutions for ordinary differential equations |
| title_exact_search | Handbook of exact solutions for ordinary differential equations |
| title_full | Handbook of exact solutions for ordinary differential equations Andrei D. Polyanin ; Valentin F. Zaitsev |
| title_fullStr | Handbook of exact solutions for ordinary differential equations Andrei D. Polyanin ; Valentin F. Zaitsev |
| title_full_unstemmed | Handbook of exact solutions for ordinary differential equations Andrei D. Polyanin ; Valentin F. Zaitsev |
| title_short | Handbook of exact solutions for ordinary differential equations |
| title_sort | handbook of exact solutions for ordinary differential equations |
| topic | Gewone differentiaalvergelijkingen gtt Oplossingen (wiskunde) gtt Équations différentielles - Solutions numériques Differential equations Numerical solutions Exakte Lösung (DE-588)4348289-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Gewone differentiaalvergelijkingen Oplossingen (wiskunde) Équations différentielles - Solutions numériques Differential equations Numerical solutions Exakte Lösung Gewöhnliche Differentialgleichung Formelsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009892069&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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