A Treatise on Differential EquationsMacmillan, 1865 - 496 páginas |
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Página 60
... system of ordi- nary differential equations dx dy dz dx_dy = P Q = Ꭱ ( 3 ) , it is evident that we shall obtain a solution of the partial dif- ferential equation ( 2 ) by constructing the system of ordinary differential equations ( 3 ) ...
... system of ordi- nary differential equations dx dy dz dx_dy = P Q = Ꭱ ( 3 ) , it is evident that we shall obtain a solution of the partial dif- ferential equation ( 2 ) by constructing the system of ordinary differential equations ( 3 ) ...
Página 61
... form the system of ordinary differential equations dx = P dy Q = dz R deduce their general integrals in the form u = a , v = b , and construct the equation F ( u , v ) = 0 . This will be the general solution sought . [ Art . 3 was ...
... form the system of ordinary differential equations dx = P dy Q = dz R deduce their general integrals in the form u = a , v = b , and construct the equation F ( u , v ) = 0 . This will be the general solution sought . [ Art . 3 was ...
Página 65
... form of the function F. I have , however , given the two proofs , chiefly in order to illustrate an important remark , viz ... system of ordinary differential equations ( 2 ) a solution of the given partial differential equation will be ...
... form of the function F. I have , however , given the two proofs , chiefly in order to illustrate an important remark , viz ... system of ordinary differential equations ( 2 ) a solution of the given partial differential equation will be ...
Página 66
... method of solution leads . That method may therefore be stated in the following Rule . RULE . To integrate the linear partial differential equation dz dx1 dz + X 2 dx z dz ... + Xn = = R dx form the system of ordinary differential ...
... method of solution leads . That method may therefore be stated in the following Rule . RULE . To integrate the linear partial differential equation dz dx1 dz + X 2 dx z dz ... + Xn = = R dx form the system of ordinary differential ...
Página 74
... SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS . 1. THE term simultaneous is here applied to a system of partial differential equations , to signify that in that system there is but one dependent variable , the general expression of which ...
... SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS . 1. THE term simultaneous is here applied to a system of partial differential equations , to signify that in that system there is but one dependent variable , the general expression of which ...
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Términos y frases comunes
¹º arbitrary constants Calculus of Variations Chap Chapter common integral complete primitive condition Crelle's Journal deduce derived determine dF dF dF dF dp dF dx dFdF differential coefficients dp dF dp dp dx dp dq dp dy dq dp du₁ du₂ dv dv dx dp dp dx dx dx dy dy dx dz dx₁ dx² dy dp dy dq dy dx dy dz dz dx eliminate equa exact differential expression factor given equation Hence homogeneous function independent variables infinite Jacobi Last Multiplier linear partial differential m₁ memoir obtain ordinary differential equations P₁ partial differential equations particular integral Professor Boole proposition reduced represent respect result shewn singular solution suppose system of ordinary theorem theory tion transformation u₁ u₂ values vanish whence x₁ y₁ аф
Pasajes populares
Página viii - On Simultaneous Differential Equations of the First Order in which the Number of the Variables exceeds by more than one the Number of the Equations, 1862, pages 437. ..454. On the Differential 'Equations of Dynamics. A sequel to a Paper on Simultaneous Differential Equations, 1863, pages 485.
Página 146 - ... second degree ; 2ndly that this system may be resolved into four systems, each consisting of two partial differential equations of the first order and first degree, two of which systems are irrelevant and the other two relevant ; Srdly that the solution of the two relevant systems ultimately depends on the solution of a system of ordinary differential equations of the first order, and that from these ordinary differential equations the given equation of the second order may be deduced independently...
Página 74 - On simultaneous differential equations of the first order in which the number of the variables exceeds by more than one the number of the equations,