A Treatise on Differential EquationsMacmillan, 1865 - 496 páginas |
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Página 3
... respect to m1 , m2 , we may affirm that the numerator is the same function of m , as the denominator is of m2 . Representing these func- tions by ( m ) , ( m , ) respectively , we have u = $ ( m , ) $ ( m . ) But m1 , m , being roots of ...
... respect to m1 , m2 , we may affirm that the numerator is the same function of m , as the denominator is of m2 . Representing these func- tions by ( m ) , ( m , ) respectively , we have u = $ ( m , ) $ ( m . ) But m1 , m , being roots of ...
Página 5
... Σ ƒn ( X , Y2 ) ( − x ̧ ) " — * + & c . , ( 32 ) ........ 1.2 ... ( nr ) " the summation extending from nr to n = ∞o . Forming hence the differential coefficients of A , with respect to ART . 5. ] 5 ADDITIONS TO CHAPTER II .
... Σ ƒn ( X , Y2 ) ( − x ̧ ) " — * + & c . , ( 32 ) ........ 1.2 ... ( nr ) " the summation extending from nr to n = ∞o . Forming hence the differential coefficients of A , with respect to ART . 5. ] 5 ADDITIONS TO CHAPTER II .
Página 6
George Boole. hence the differential coefficients of A , with respect to x yo , and reducing by ( 28 ) , we shall find and dA , dA . + f1 ( xo , Yo ) = = 0 , dx dy 。 whence in particular dA dA 0 dx . + f1 ( x 。, Yo ) = 0 . dy Eliminate ...
George Boole. hence the differential coefficients of A , with respect to x yo , and reducing by ( 28 ) , we shall find and dA , dA . + f1 ( xo , Yo ) = = 0 , dx dy 。 whence in particular dA dA 0 dx . + f1 ( x 。, Yo ) = 0 . dy Eliminate ...
Página 26
... respect to dx x , on the above hypothesis as to the constitution of v as a function of x and u , while is the differential coefficient ( dv dec on the hypothesis that v is reduced to a function of x alone by the conversion of u into a ...
... respect to dx x , on the above hypothesis as to the constitution of v as a function of x and u , while is the differential coefficient ( dv dec on the hypothesis that v is reduced to a function of x alone by the conversion of u into a ...
Página 27
... respect to x . Hence the equation re- duces to 0 = Q. And this is not satisfied , since by hypothesis Q does not ... respecting Q , and gives U = f ( x , u ) , we see that if du dx = f ( x , u ) be satisfied by u = 0 , and if at the same ...
... respect to x . Hence the equation re- duces to 0 = Q. And this is not satisfied , since by hypothesis Q does not ... respecting Q , and gives U = f ( x , u ) , we see that if du dx = f ( x , u ) be satisfied by u = 0 , and if at the same ...
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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,