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X6n-4 = h X6n-3 = n-1 f3i+29-f3ik h II (J3i+1p-fat-sh) (136429 - fak) f3i-1p-f3i-3h, i=0 X6n-2 = n-1 k II i=0 (f3i+3P - f3i+1h) f3i+1P-f3i-1h, - - f3i+19f3i-1k` f3i-19-f3i-3k (3) n-1 - II (fai+2P - fash) (fai+39 - fai+1k) - i=0

X6n-4 = h X6n-3 = n-1 f3i+29-f3ik h II (J3i+1p-fat-sh) (136429 - fak) f3i-1p-f3i-3h, i=0 X6n-2 = n-1 k II i=0 (f3i+3P - f3i+1h) f3i+1P-f3i-1h, - - f3i+19f3i-1k` f3i-19-f3i-3k (3) n-1 - II (fai+2P - fash) (fai+39 - fai+1k) - i=0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 5. Let {xn}4 be a solution of Eq.(9). Then for n = 0, 1, 2, ... n=-4 n-1 f3i+1P – f3i-1h` f3i-1p – fai-zh) \Fas9 – fai-2k ) ' fai+29 – fzik\ » X6n-4 i=0 п-1 fai+3P – fai+ih\ ( fai+19 – f3i–1k) Fsi+1P – fai-1h) (Fsi-19 – f3i-3k , X6n-3 i=0 n-1 ( fai+39 – fai+1k П X6n-2 f3iP – fai-zh) i=0 PI(si+4P – f3i+2h) fai+2P – faih п-1 ( f3i+29 – f3ik \f3:9 – fzi-2k ) X6n-1 i=0 п-1 fai+3P – fsi+1h \ f3i+1P – f3i-1h, (fsi+49 – fsi+2k fzi+29 – f3ik X6n i=0 п-1 ( fzi+5P – fzi+3h` П f3i+3P – fai+1h, fai+39 – f3i+1k` fsi+19 – fai-1k ) 2р — h X6n+1 р — h i=0 wherer-4 = h, x_3 = k, x_2 = r, x_1 = P, x0 = q, {fm}m=-1 = {-1, 1, 0, 1, 1, 2, 3, 5, 8, ...}.
Transcribed Image Text:Theorem 5. Let {xn}4 be a solution of Eq.(9). Then for n = 0, 1, 2, ... n=-4 n-1 f3i+1P – f3i-1h` f3i-1p – fai-zh) \Fas9 – fai-2k ) ' fai+29 – fzik\ » X6n-4 i=0 п-1 fai+3P – fai+ih\ ( fai+19 – f3i–1k) Fsi+1P – fai-1h) (Fsi-19 – f3i-3k , X6n-3 i=0 n-1 ( fai+39 – fai+1k П X6n-2 f3iP – fai-zh) i=0 PI(si+4P – f3i+2h) fai+2P – faih п-1 ( f3i+29 – f3ik \f3:9 – fzi-2k ) X6n-1 i=0 п-1 fai+3P – fsi+1h \ f3i+1P – f3i-1h, (fsi+49 – fsi+2k fzi+29 – f3ik X6n i=0 п-1 ( fzi+5P – fzi+3h` П f3i+3P – fai+1h, fai+39 – f3i+1k` fsi+19 – fai-1k ) 2р — h X6n+1 р — h i=0 wherer-4 = h, x_3 = k, x_2 = r, x_1 = P, x0 = q, {fm}m=-1 = {-1, 1, 0, 1, 1, 2, 3, 5, 8, ...}.
Bxn-1n-2 YIn-1 + 8xn-4 Xn+1 = axn-2 + n = 0, 1, .., (1) A specific form of the solutions of the difference equation has been provided Xn-1Xn-2 Xn+1 = xn-2+ (9) Xn-1 - Xn-4 where the initial conditions x-4, x_3, x-2, x-1, xo are arbitrary non zero real num- bers with x_4 7 x-1-
Transcribed Image Text:Bxn-1n-2 YIn-1 + 8xn-4 Xn+1 = axn-2 + n = 0, 1, .., (1) A specific form of the solutions of the difference equation has been provided Xn-1Xn-2 Xn+1 = xn-2+ (9) Xn-1 - Xn-4 where the initial conditions x-4, x_3, x-2, x-1, xo are arbitrary non zero real num- bers with x_4 7 x-1-
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