A Treatise on the Calculus of Finite Differences |
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Page 39
... replacing Au , by u1 - u , and so on , we find , reduction , on Ju dx 3 = 10 { u + u2 + u ̧ + u ̧ +5 ( u ̧ + u ̧ ) + 6u ̧ } , 0 which , supposing the common distance of the ordinates to be h , gives 6h [ ou_dx = = 3h 10 { μ ̧ + μ¿ + u ̧ ...
... replacing Au , by u1 - u , and so on , we find , reduction , on Ju dx 3 = 10 { u + u2 + u ̧ + u ̧ +5 ( u ̧ + u ̧ ) + 6u ̧ } , 0 which , supposing the common distance of the ordinates to be h , gives 6h [ ou_dx = = 3h 10 { μ ̧ + μ¿ + u ̧ ...
Page 61
... replacing of that question by others . But the answers will not be of necessity sufficiently general , and the substituted questions if answered in a perfectly un- restricted manner may lead to results which are too general . In the one ...
... replacing of that question by others . But the answers will not be of necessity sufficiently general , and the substituted questions if answered in a perfectly un- restricted manner may lead to results which are too general . In the one ...
Page 81
... replacing t by dx and restoring Συχ Σu2 = ( d ) ~ the subject u , we have -1 Их - 1 1 dux 1 d3ux ux + 12 dx 720 dx3 + & c . = C + fu , dx —— 1 ux + 2 1 du 1 d3uz - 12 dx 720 da + & c ....... ( 2 ) . For most applications this would ...
... replacing t by dx and restoring Συχ Σu2 = ( d ) ~ the subject u , we have -1 Их - 1 1 dux 1 d3ux ux + 12 dx 720 dx3 + & c . = C + fu , dx —— 1 ux + 2 1 du 1 d3uz - 12 dx 720 da + & c ....... ( 2 ) . For most applications this would ...
Page 103
... replacing C by a variable para- meter C , and then , in analogy with the known method of solution for linear differential equations , seek to deter- mine C. We have U2 = С2PAα - 11 Ux + 1 = Cx + 1 PA2 whence ( 1 ) becomes But whence or ...
... replacing C by a variable para- meter C , and then , in analogy with the known method of solution for linear differential equations , seek to deter- mine C. We have U2 = С2PAα - 11 Ux + 1 = Cx + 1 PA2 whence ( 1 ) becomes But whence or ...
Page 105
... Replacing Au , by ux + 1- ux , we have Ux + 1 + U2 = − ( x + 1 ) . Here A - 1 , B2 = − ( x + 1 ) , whence substituting in ( 7 ) and reducing Ux = C ( − 1 ) * . Ex . 4 . Ux + 1 = aux = We find Ux = ax - 1 = α x - 1 - x 1 2 ax ( x + 1 ) ...
... Replacing Au , by ux + 1- ux , we have Ux + 1 + U2 = − ( x + 1 ) . Here A - 1 , B2 = − ( x + 1 ) , whence substituting in ( 7 ) and reducing Ux = C ( − 1 ) * . Ex . 4 . Ux + 1 = aux = We find Ux = ax - 1 = α x - 1 - x 1 2 ax ( x + 1 ) ...
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Common terms and phrases
A₁ A³u algebraic applying arbitrary constant ascending powers assume ax+b B₁ c₂ Calculus of Finite Cambridge CHARLES KINGSLEY cloth CO.'S PUBLICATIONS complete primitive CONIC SECTIONS convergent Crown 8vo curve deduce denoted determine Differential Calculus differential equation divergent English Notes equation of differences expression Fcap Fellow of St Finite Differences functional equation GEORGE BOOLE given equation Hence independent variable indirect integral John's College limit linear equation M.A. Second Edition MACMILLAN & CO.'S method mixed differences operation ordinates P. G. TAIT particular value periodical function rational and integral reduced result Schools second member Sermons shew singular solution substituting successive values supposed symbol theorem tion TODHUNTER Treatise Trinity College u₁ u₂ unity University of Cambridge Ux+1 Ux+2 v₂ Vx+1 Vx+2 whence Их
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