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| Авторизация |
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| Goursat E. — A course in mathematical analysis. Volume 2, part 2: Differential equations |
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| Предметный указатель |
Homogeneous linear equations, invariants 445 41
Homogeneous linear equations, Lamp's equation 146 63
Homogeneous linear equations, Laplace's equation 124
Homogeneous linear equations, Legendre's equation 112
Homogeneous linear equations, linearly Independent integrals 405 405 88
Homogeneous linear equations, periodic coefficients 128 47; 64; ex.;
Homogeneous linear equations, permutations of integrals around a critical point 129 47
Homogeneous linear equations, Picard's equation 146 63
Homogeneous linear equations, ratio of two integrals 169 ex.
Homogeneous linear equations, regular 454 60
Homogeneous linear equations, regular integrals 128 47; 47; 49
Homogeneous linear equations, relation to Riccati's equation 111 40;
Homogeneous linear equations, roots of integrals, Sturm's theorem 111
Homogeneous linear equations, solution as a definite integral see "Definite integrals"
Homogeneous linear equations, system of see "System of homogeneous linear equations"
Homogeneous linear equations, Wronskian 129 47; "Characteristic
Hyperelliptic functions 405 68
Hypergeometric series 440 51
Hypergeometric series degenerate cases 142 52
Identical transformation 88 34; 36
Incompressible fluid, condition for 86 33
Incompressible fluid, invariant integrals 84 33
Independent equations 265 88
Independent integrals 81 31
Independent integrals, linearly 405 38; 38
Infinitesimal transformations 86 34; 36; 36; 36
Initial conditions 45 22; 22; 22; 23; 24; 24; 26;
Initial conditions, defining an integral 37
Initial Conditions, Partial Differential Equations 54 25; 25; 74; 76; 84; 94; "Derivatives
Integra equation 64 27
Integrable combination 77 31; exs. 2; 76; 83
Integral curves 4 1; 26 26; Note 65; 71
Integral curves in parametric form 46 9
Integral curves of a partial differential equation 257 86; ex. ex.
Integral curves, center 450 65
Integral curves, cusps see "Cusps"
Integral curves, envelope of 47 10; 71; 72; 74; ex.
Integral curves, focus 180 65
Integral curves, saddleback 470 65;
Integral surfaces 218 76; 76; 78; 84; 85; 86;
Integral surfaces, envelope of 288 82; 93;
Integrals, Abelian 18 11
Integrals, analytic see "Analytic extension" and "Analytic integrals"
Integrals, anharmonic ratio of 18 7
Integrals, Cauchy's problem 246 264 87
Integrals, common to two linear equations 114 41
Integrals, complete see "Complete integral"
Integrals, defined by initial conditions 100 37
Integrals, elements of 251 86; 87
Integrals, elliptic functions 19 ex. 20; 63; 68
Integrals, existence of see "Existence theorems"
Integrals, first see "First integrals"
Integrals, fundamental system of see "Fundamental system of integrals"
Integrals, general see "General integral"
Integrals, general properties of 100 37
Integrals, hypergeometric series 140 61; 62;
Integrals, independent 81 31
Integrals, initial conditions see "Initial conditions"
Integrals, invariant see "Invariant integrals"
Integrals, Legendre's polynomials 112 ex.
Integrals, Lie's enlarged definition 264 Note
Integrals, linearly independent 103 38; 38
Integrals, non-analytic see "Noilanalytic integrals"
Integrals, of equations of higher order 196 70
Integrals, particular 3 1; 7; 7; 12; 39; 40
Integrals, periodic 192 68
Integrals, permutation of integrals around a critical point 129 47; 49
Integrals, rational functions 144 63; 68
Integrals, rational functions of constants 10 4; 186 67
Integrals, regular 128 47; 47; 49
Integrals, roots of, Sturm's theorem 111
Integrals, singular see "Singular integrals"
Integrals, singular points see "Singular points"
Integrals, Wronskian 129 47; "Integral "Integral
Integrating factors 119 12; 32; 32; 36; exs. 4; 42; 80; ex.
Integrating factors, existence of 57 26
Integration by raising order 41 Note
Invariance of condition of integrability 281 80
Invariant functions 98 36
Invariant integral 88 33
Invariant integral of homogeneous linear equations 115 41
Invariant integral of incompressible fluid 84 33
Invariant integral, line and surface 84 63
Invariant integral, multiple 85 33
Invariant integral, volume 86 33
Involutory systems 274 91
Involutory systems, complete integral 277 91
Involutory systems, Jacobi's method 277 92
Involutory systems, Poisson's parenthesis 274 91
Isothermal curves 48 ex.
jacobi 11 6; 14; 16; 31; 32; 60; 89; 89; ex.; 91; 92; 92
Jacobi's equation 11 6; 16
Jacobi's equation, relation to a system of homogeneous linear equations 168 60
Jacobi's method, involutory systems 277 92
Jacobi's multipliers 74 31; 32
Jacobian system 269 89; 89; ex.; 91; 92
Kovalevsky, Madame 45 21; 94
Lagrange 16 9; ex. 39; 42; 71; 72; 72; ex. 82; 82; 83; 83; 85; ex. 86; 92
Lagrange and Charpit's method 240 83; 92
Lagrange's equation 16 9; 72; 72
Lagrange's integral of Euler's equation 48 ex.
Lagrange's method 241 83; 86
Lagrange's method of the variation of constants 107 39;
Lagrange's theory of the complete integral 286 82; 82; ex. 86
Laguerre 115 41
lame 146 63
Lame's equation 146 53
Laplace 124 40; Note
Laplace's linear equation 124 40; Note
legendre 16; 15 15; ex.
Legendre's 16
Legendre's equation 112 ex.
Legendre's of linear equations 115 41; 59
Legendre's of Riccati's equation 15; see also "Groups and Invariants"
Legendre's polynomials 112 ex.
Legendre's transformation 16
Leibnitz 118 43
Leibnitz's formula 118 43
Liapunof 151 55 62
Lie 45 ex. 30; Note; ex.
Lie's enlarged definition of the integral 264 Note
Lie's theory of differential equations 86 34;
Lindelof 61 27; ex.
Linear equations 9 4; 35; 37; 39; 67
Linear equations, coefficients depending upon a parameter 65 Note
Linear equations, depression of order 109 40
Linear equations, general properties of integrals 100 37; "Integrals" "Non-homogeneous "Partial
Linearly independent functions 103 38
Linearly independent functions, integrals 103 38; 38
Lines of an ellipsoid 41 Note
Lines of curvature 206 72
Lines of curvature of a helicoid 91 35
Liouville 79 ex.
Liouville's equation 79 ex.
Lipschitz 68 30; 94
Lipschitz condition 68 30; 94
Locus, of characteristic curves 219 76
Locus, of cusps of integral curves 201 71; 71; 72; 73; 74; ex.
Locus, of points of inflection of integral curves 213 ex.
Mayer 229 79
Mayer's method 229 79
Moigno 68 30; ex.
Monge 41 Note
Monge's method of finding the lines of curvature of an ellipsoid 41 Note
Movable singular points 181 00; 67; 70
Movable singular points for equations of higher order 196 70
Movable singular points, essentially singular 196 70
Movable singular points, lines of 197 70
Movable singular points, poles 197
Movable singular points, transcendental critical points 197 70
| Mtray 45 21
Multipliers 74 31; 32; 33
Non-analytic integrals 50 22;
Non-analytic integrals, Briot and Bouquet's theorem 175 04; 64; 04; 05;
Non-homogeneous linear equations 100 37; 39
Non-homogeneous linear equations, analytic extension of integrals 101 37
Non-homogeneous linear equations, Cauchy's method 108 39;
Non-homogeneous linear equations, constant coefficients 120 43
Non-homogeneous linear equations, corresponding homogeneous equation 107 39
Non-homogeneous linear equations, depression of order 110 40
Non-homogeneous linear equations, general integral 107 39
Non-homogeneous linear equations, Lagrange's method of the variation of constants 107 109
Non-homogeneous linear equations, singular points 100
Non-homogeneous linear equations, system of equations see "Systems of non-homogeneous linear equations"
Non-linear differential equations 172 03; 65
Non-linear differential equations, (derivative indeterminate) 173 64; 65
Non-linear differential equations, (derivative infinite) 172 03
Non-linear differential equations, algebraic equations of the first order 180 66; 67
Non-linear differential equations, Briot and Bouquet's problem 193
Non-linear differential equations, exceptional initial values 172 63
Non-linear differential equations, functions defined by y' = R(x, y) 182 67
Non-linear differential equations, having single-valued integrals 187 68; 68;
Non-linear differential equations, integrals see "Envelope of integrals" "Integral "Integrals" "Locus
Non-linear differential equations, non-analytic integrals see "Non-analytic integrals"
Non-linear differential equations, singular points see "Critical points" "Fixed "Movable "Singular
Non-linear differential equations, systems of 208 74; "Clairaut" "Euler" "Lagrange"
Normal form of a system of partial differential equations 283 94
Order of a differential equation 4 1
Order of a differential equation, depression of 86 19; 40
Order of a differential equation, first order 6 2; 66
Order of a differential equation, higher order, nth order 4 1; 2; 18; 70
Order of a differential equation, integration by raising order 42 Note
Orthogonal surfaces 223 77
Orthogonal trajectories 24 7; 17; ex. 78
Painlevd's equation 200 70; 70
Painlevt 59; 74 30; 70; 70; ex.
Parallel curves 42 ex.
Parallel surfaces 289 ex.
Parenthesis (u, v) 234 81
Partial differential equations 70 31
Partial differential equations of first order see "Partial differential equations of the first order"
Partial differential equations of higher order 278 93
Partial differential equations of ruled surfaces 280 ex. ex.
Partial differential equations of the first order, linear 76 31; 76
Partial differential equations of the first order, linear of helicoids 220 ex. 83
Partial differential equations of the first order, linear, characteristic curves see "Characteristic curves"
Partial differential equations of the first order, linear, general integral 217 76
Partial differential equations of the first order, linear, general method of integration 224 76
Partial differential equations of the first order, linear, geometric interpretation 218 76
Partial differential equations of the first order, linear, initial conditions 221 76
Partial differential equations of the first order, linear, integral surface 218 76; 76
Partial differential equations of the first order, linear, of conoids 220 ex.
Partial differential equations of the first order, linear, singular integral, surface 224 77;
Partial differential equations of the first order, non-linear, (extended) 261 87
Partial differential equations of the first order, non-linear, any number of variables 261 87
Partial differential equations of the first order, non-linear, Cauchy's equation 267 ex.
Partial differential equations of the first order, non-linear, Cauchy's method 249 86; 86; Note
Partial differential equations of the first order, non-linear, Cauchy's problem 246 84
Partial differential equations of the first order, non-linear, Clairaut's equation, generalized 239 ex.
Partial differential equations of the first order, non-linear, complete integral 286 82;
Partial differential equations of the first order, non-linear, element 251 85
Partial differential equations of the first order, non-linear, envelope of surfaces 238 82
Partial differential equations of the first order, non-linear, general integral 238 82
Partial differential equations of the first order, non-linear, integral curves 257 85; ex.
Partial differential equations of the first order, non-linear, integral, Lie's enlarged definition: 264 Note
Partial differential equations of the first order, non-linear, Lagrange and Charpit's method 240 83;
Partial differential equations of the first order, non-linear, separation of variables 244 ex.
Partial differential equations of the first order, non-linear, singular integrals 224 77; 82; 90
Partial differential equations of the first order, non-linear, three variables 236 82;
Partial differential equations of tubular surfaces 240 ex. ex.;
Partial differential equations, (system of equations) 283 94
Particular integral, solution 10 1; 7; 7; 12; 39; 40
Periodic coefficients 128 47; 54; ex.;
Periodic coefficients, elliptic 244 53; 54
Picard 59; 02 27; 116 42; 53; 64
Picard's equation 244 53
Picard's equation, system of linear equations 164 61; 62
Picard's method of successive approximations see "Successive approximations"
Poincari 100 33; 64; 65;
poisson 274 91; 92
Poisson's 274 91
Poisson's identity 277 92
Poisson's parenthesis 274 91
Poles of integrals 240 53; 67; 97
Poles of integrals, movable 200 197
Properties, of  , tan x 218 ex.
Properties, of differential equations of higher order 196 70
Quadratures 7 2; 4; 7; 7; 7; 9; 70 31; Note 32; 36; 39; 40; 40; 66; 92
Raffy 44 ex.
Ratio, of similitude 0 3
Ratio, of twointegrals 169 ex.
Rational functions, of constants as integrals 20 4; 7; 67
Rational functions, of variables as integrals 244 53; 68
Reducible systems 165 62
Regular differential equations 134 50
Regular integrals 128 47; 47; 49
Regular integrals, Fuchs' theorem 254 50
Regular integrals, substitutions 132 48; 48
Riccati 12 7; ex. ex. 40; 40 Note; 57; ex. ex. 67; 67 ex.
Riccati's equation 25 7; ex. ex. 40; 40 Note; 57; ex. ex. 67 67; ex.
Riccati's equation, generalization of 297
Riccati's equation, linear transformation of 25
Riccati's equation, relation to linear equations 111 40;
Riquier 45 21; 94
Roots of characteristic equation 130 47; 48; 50; Note
Roots of characteristic equation, elementary divisors 132
Roots of integrals Sturm's theorem 111
Ruled surfaces 280 ex. ex.
Saddleback 279 65
Sauvage 132
Schloemilch 212 ex.
Separation of variables 6 2; 3; 12; ex.
Serret 212 ex. ex.
Similar curves 5 3
Similar groups 55 34
Single-valued integrals 144 53;
Single-valued integrals of  , classification of equations 257 68; 68;
Singular integral as an envelope 203 71; 82
Singular integral, curve, surface 27 10; 14; 71; 71; 72; 72; 74; 74; 77; 82;
Singular integral, curves and surfaces determination of 205 72
Singular integral, curves and surfaces determination of, geometric interpretation 207 73
Singular integral, curves and surfaces determination of, of a system of equations 208 74; 74
Singular integral, curves and surfaces determination of, of firstorder equations 198 71; 71; 72
Singular integral, curves and surfaces determination of, of partial differential equations 224 77; 82; 90
Singular integral, geometric interpretation 207 73
Singular lines, movable 297 70
Singular points, algebraic critical points 173 63; 67; 67; 71
Singular points, Briot and Bouquet's theorem 276 64
Singular points, center 180 65
Singular points, essentially 252 47; 49
Singular points, essentially singular movable 196 70
Singular points, fixed 181 66; 67
Singular points, focus 130 65
Singular points, indeterminate derivative 275 64
Singular points, infinite derivative 172 63
Singular points, infinite number of critical points 185
Singular points, movable see "Singular points" "Movable
Singular points, of equations of the first order 180 66
Singular points, of linear equations 65 28; 37; 47; 51; 52; 53
Singular points, poles 252 47; 53; 53; 67; 67; 67;
Singular points, saddleback 179 65
Solution see "Integral"
Star 67 29
Stationary flow 86 33
Stieltjes 27 14
Stieltjes's general integral of Euler's equation 27 14
Straight lines, differential equation of 4 1
Sturm 111
Sturm's theorem 111
Substitutions linear equations 129 47; 48; 48
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