Partial Differential Equations: An Essay Towards an Entirely New Method of Integrating Them |
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Page 17
... separately with D , and A ; and also multiply it by b ; the results are ( since D and At are commutative ) D¿¿ . D¿u + b Dr. D¿u = 0 , Dƒat · A¿u + bD ̧ . △ ¡ μ = 0 , DA . bu + bD1 . bu = 0 ; · . :: Dƒa¿ . ( D¿u + △ u + bu ) + bD1 ...
... separately with D , and A ; and also multiply it by b ; the results are ( since D and At are commutative ) D¿¿ . D¿u + b Dr. D¿u = 0 , Dƒat · A¿u + bD ̧ . △ ¡ μ = 0 , DA . bu + bD1 . bu = 0 ; · . :: Dƒa¿ . ( D¿u + △ u + bu ) + bD1 ...
Page 18
... separately ( should it be advantageous to do so ) ; and their sum will constitute the complete integral . The absolute part corresponds to the general supposition , W = 0 . As an example of this principle , let it be required to inte ...
... separately ( should it be advantageous to do so ) ; and their sum will constitute the complete integral . The absolute part corresponds to the general supposition , W = 0 . As an example of this principle , let it be required to inte ...
Page 19
... separately with sm , St , Σt ; the results are .. Dr △ fu = sm0 , dm △ ? u = S * 0 , d * D * u = Σf O. Add these equations together , writing u for the sum of their left - hand members ; .. u = sm0 + SO + Σ { O . Now this value of u ...
... separately with sm , St , Σt ; the results are .. Dr △ fu = sm0 , dm △ ? u = S * 0 , d * D * u = Σf O. Add these equations together , writing u for the sum of their left - hand members ; .. u = sm0 + SO + Σ { O . Now this value of u ...
Page 38
... separately with St , Et , 8 ,; and it will involve four arbitrary functions , F ( k , y ) , f ( ' , Y ) , đ , ( , Ë ) , Ô , ( Ë , ŕ ' ) . But the following integral in finite terms , though in- volving only two arbitrary functions , may ...
... separately with St , Et , 8 ,; and it will involve four arbitrary functions , F ( k , y ) , f ( ' , Y ) , đ , ( , Ë ) , Ô , ( Ë , ŕ ' ) . But the following integral in finite terms , though in- volving only two arbitrary functions , may ...
Page 53
... separately and similarly . 1 1 — 8 ; 0 = ( 4 , + 4 ̧t + 4 , ť2 + ... + Â1⁄2‚t * ́1 ) t = = Âť1 + à ̧ +  ̧t + ... + A x_th2 , 1 A。, A1 , 4 , ... being functions of § . 1 .. Dr1 . — S ‡ 0 = F ' ( § ) . t * ; t .. ( DA ) ~ -Σ ; 0 ...
... separately and similarly . 1 1 — 8 ; 0 = ( 4 , + 4 ̧t + 4 , ť2 + ... + Â1⁄2‚t * ́1 ) t = = Âť1 + à ̧ +  ̧t + ... + A x_th2 , 1 A。, A1 , 4 , ... being functions of § . 1 .. Dr1 . — S ‡ 0 = F ' ( § ) . t * ; t .. ( DA ) ~ -Σ ; 0 ...
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A₁ ad¸u Algebra arbitrary constants arbitrary functions ARITHMETIC beginners BROOKE FOSS WESTCOTT Cambridge Chapter CLASS-BOOK cloth coefficients commutative complete integral Crown 8vo d¸³u D₁ D₁u D₁x D₂ d³u Differential Calculus EDWARD THRING ELEMENTARY TREATISE English equa equivalent Essay Examination Papers Extra fcap F₁ Fcap final integral finite terms form of H ƒ Dr ƒ dt GEOMETRY gives Grammar Greek ht+kx independent variables integral in finite John's College kx)² Latin liberty to assume Mathematical method MIXED MATHEMATICS Multiply numerous Examples obtained partial differential equation preceding example principle proposed equation quasi-constants render revised root-integral Schools Second Edition second order shew student substitute symbolical integral symbols of operation tion Trigonometry utª write
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