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J6i+3P+J6i+2" ) J6i+54+J6i+4/ fei+4P+ fei+3h foi+3P+ fei+2h) ( foi+59 + foitak ) fei+64+ fei+5k = q II i=0 п-2 () + - IT (foi+2p+fei+1h foi+sp+f6i+7h i=0 )(4 foi+7p+ f6i+6h foi+1p+f6ih i=0 2p+h p+h П n-2 foi+4p+ f6i+3h 2p+hg| foi+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k fei+69 + foi+sk fei+4P+ fei+3") (i + fei+ak) n-2 p+h i=0 fei+3P + f6i+2h i=0 foi+59 + fei+ak, () + () ( 2p+h p+h 2p+h p+h fon-5p+fen-6h fon-4p+fen-5h n-2 Sei+3p+fei+2h) i=0 fGi+69+S6i+5k Soi+59+S6i+4k n-2 foi+4P + foi+3h ( foi+69 + föi+sk П foi+3p + foi+2h foi+59 + fei+ak ) fen-5p+ fen-6h 1+ i=0 fon-4p+ fon-5h n-2 П foi+4P+foi+3h f6i+3P+f6i+2h ( fei+69+S6i +5k n-2 Soi+59+S6i+4k -T( foi+4P + f6i+3h foi+69+ fei+sk «П fei+3p + fei+2h ) ( ) foi+59 + foi+ak fön-4p+fon-sh+fen-5p+fon-6h fon-4P+ fon-5h n-2 f6i+4p+f6i+3h foi+3p+ fei+2h (föi+69+f6i +5k f6i+59+f6i+4k fei+69 + fei+5k foi+4P + foi+3" ) fau54 + feitak / n-2 = «II ( i=0 foi+3P + fei+2h, foi+59 + foi+ak, fon-3p+ fen-4h fon-4p+ fên-sh i=0 n-2 ( fei+4P + föi+3h\ ( foi+64 + foi+5k föi+3p + foi+2h) foi+69+ fei+5k foi+39 + fei+ak ) [* + (Fon-3p + fon-ah )] fön-4p+ fen-5h = q II i=0

J6i+3P+J6i+2" ) J6i+54+J6i+4/ fei+4P+ fei+3h foi+3P+ fei+2h) ( foi+59 + foitak ) fei+64+ fei+5k = q II i=0 п-2 () + - IT (foi+2p+fei+1h foi+sp+f6i+7h i=0 )(4 foi+7p+ f6i+6h foi+1p+f6ih i=0 2p+h p+h П n-2 foi+4p+ f6i+3h 2p+hg| foi+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k fei+69 + foi+sk fei+4P+ fei+3") (i + fei+ak) n-2 p+h i=0 fei+3P + f6i+2h i=0 foi+59 + fei+ak, () + () ( 2p+h p+h 2p+h p+h fon-5p+fen-6h fon-4p+fen-5h n-2 Sei+3p+fei+2h) i=0 fGi+69+S6i+5k Soi+59+S6i+4k n-2 foi+4P + foi+3h ( foi+69 + föi+sk П foi+3p + foi+2h foi+59 + fei+ak ) fen-5p+ fen-6h 1+ i=0 fon-4p+ fon-5h n-2 П foi+4P+foi+3h f6i+3P+f6i+2h ( fei+69+S6i +5k n-2 Soi+59+S6i+4k -T( foi+4P + f6i+3h foi+69+ fei+sk «П fei+3p + fei+2h ) ( ) foi+59 + foi+ak fön-4p+fon-sh+fen-5p+fon-6h fon-4P+ fon-5h n-2 f6i+4p+f6i+3h foi+3p+ fei+2h (föi+69+f6i +5k f6i+59+f6i+4k fei+69 + fei+5k foi+4P + foi+3" ) fau54 + feitak / n-2 = «II ( i=0 foi+3P + fei+2h, foi+59 + foi+ak, fon-3p+ fen-4h fon-4p+ fên-sh i=0 n-2 ( fei+4P + föi+3h\ ( foi+64 + foi+5k föi+3p + foi+2h) foi+69+ fei+5k foi+39 + fei+ak ) [* + (Fon-3p + fon-ah )] fön-4p+ fen-5h = q II 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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show me steps of the determine green and all information is here explain to me step by step i need every details

Bxn-10n-2 yXn-1 + 8xn-4 In+1 = aXn-2 + n = 0, 1, ..., (1) 1 The following special case of Eq.(1) has been studied In-1*n-2 Tn+1 = Xn-2 + (8) Xn-1+ xn-4 where the initial conditions -4, x-3, x-2, x-1,and xo are arbitrary non zero real numbers. Theorem 4. Let {rn}-4 be a solution of Eq.(8). Then for n = 0, 1, 2, ... Therefore n-1 foip + fei-ih fei+29 + f6i+1k` hII foi-1p+ foi-2h ) foi+19 + feik X6n-4 = h i=0 Also, from Eq.(8), we see that X6n-5X6n-6 X6n-3 = X6n-6 + X6n-5 + x6n-8 n-2 fei+69+ f6i+5k foi+59 + föi+ak fei+4P + f6i+3h 9II foi+3P + fei+2h, i=0 n-2 n-2 Sei+8P+S6i+7h(f6i+49+f6i+3* [(Tei+3P+S6i+2h)(Tei+59+S6i+4k i=0 i=0 n-2 n-2 +r \Tôi+39+S6i+2k / Toi+1p+/6ih )(Toi+39+S6i+2k, i=0 i=0 13 n-2 fei+4P + fei+3h «II fsi+3P + f6i+2h foi+69+ fei+sk fei+59 + foi+ak, i=0 n-2 2p+h pth Joi+3p+fei+2h feit69+Sei+sk Jei+59+Sei+ak i=0 п-2 п-2 n-2 f6i+sp+fei+7h 2p+h p+h) 11 (Tei+7p+f6i+sh fsi+7p+S6i+6h S6i+1p+S6ih i=0 i=0 i=0 n-2 2p+ha TI (fsit+4P+Ssi+3h f6i+3P+f6i+2h fsi+69+Ssi+sk ( Tôi+59+S6i+ak п-2 fei+69 + fei+5k foi+39 + f6i+ak, fei+aP + f6i+3h` qII i=0 = q foi+3P + fei+2h п-2 2p+h p+h fei+2p+ f6i+1h II (Toi+sp+f6i +7h, fsi+7p+fei+ch Ssi+1p+feih i=0 i=0 n-2 2p+h aTI (fsi+4P+S6i+3h Si+3p+f6i+2h Sei+69+f6i+5k J6i+59+S6i+ak п-2 fei+4P + fei+3h fei+3P + f6i+2h ) foi+69 + fei+sk` f6i+59 + fei+ak, i=0 = qII () + () ( ) 2p+h pth 2p+h p+h fên-5p+S6n-6h fên –4p+f6n-5h i=0 n-2 Sei+69+f6i+5k Soi+s9+f6i+4k п-2 foi+3p+ f6i+2h) foi+ap + fei+3h foi+3p + foi+2h foi+69+ fei+sk + i=0 = q foi+59 + f6i+ak, 1+ fon-sp+Sên-6h Sên-4p+ fén-sh i=0 n-2 feit4p+S6i+3h Jei+59+Sei+ak = qT(föi+4P + foi+3h fei+3P + föi+2h n-2 foi+69 + f6i+5k` fei+59 + fei+ak i=0 fon-4p+fon-sh+fén-5p+fën-6h Sên-4P+Sên-sh i=0 п-2 Soi+69+S6i+5k Sei+59+Sei+4k п-2 fei+3p+f6i+2h fei+4P+ fei+3h\ ( foi+69 + fei+5k + i=0 = q|| foi+3P + fei+2h) foi+59 + f6i+ak, fên-3p+ fen–4h Sen-4p+ fen–sh i=0 n-2 (foi+4p+ fei+3h f6i+3p + fsi+2h , foi+6q+ fei+sk foi+5q + fei+ak (fön-4p + fön-sh fön-3p + fön-4h i=0 14
Transcribed Image Text:Bxn-10n-2 yXn-1 + 8xn-4 In+1 = aXn-2 + n = 0, 1, ..., (1) 1 The following special case of Eq.(1) has been studied In-1*n-2 Tn+1 = Xn-2 + (8) Xn-1+ xn-4 where the initial conditions -4, x-3, x-2, x-1,and xo are arbitrary non zero real numbers. Theorem 4. Let {rn}-4 be a solution of Eq.(8). Then for n = 0, 1, 2, ... Therefore n-1 foip + fei-ih fei+29 + f6i+1k` hII foi-1p+ foi-2h ) foi+19 + feik X6n-4 = h i=0 Also, from Eq.(8), we see that X6n-5X6n-6 X6n-3 = X6n-6 + X6n-5 + x6n-8 n-2 fei+69+ f6i+5k foi+59 + föi+ak fei+4P + f6i+3h 9II foi+3P + fei+2h, i=0 n-2 n-2 Sei+8P+S6i+7h(f6i+49+f6i+3* [(Tei+3P+S6i+2h)(Tei+59+S6i+4k i=0 i=0 n-2 n-2 +r \Tôi+39+S6i+2k / Toi+1p+/6ih )(Toi+39+S6i+2k, i=0 i=0 13 n-2 fei+4P + fei+3h «II fsi+3P + f6i+2h foi+69+ fei+sk fei+59 + foi+ak, i=0 n-2 2p+h pth Joi+3p+fei+2h feit69+Sei+sk Jei+59+Sei+ak i=0 п-2 п-2 n-2 f6i+sp+fei+7h 2p+h p+h) 11 (Tei+7p+f6i+sh fsi+7p+S6i+6h S6i+1p+S6ih i=0 i=0 i=0 n-2 2p+ha TI (fsit+4P+Ssi+3h f6i+3P+f6i+2h fsi+69+Ssi+sk ( Tôi+59+S6i+ak п-2 fei+69 + fei+5k foi+39 + f6i+ak, fei+aP + f6i+3h` qII i=0 = q foi+3P + fei+2h п-2 2p+h p+h fei+2p+ f6i+1h II (Toi+sp+f6i +7h, fsi+7p+fei+ch Ssi+1p+feih i=0 i=0 n-2 2p+h aTI (fsi+4P+S6i+3h Si+3p+f6i+2h Sei+69+f6i+5k J6i+59+S6i+ak п-2 fei+4P + fei+3h fei+3P + f6i+2h ) foi+69 + fei+sk` f6i+59 + fei+ak, i=0 = qII () + () ( ) 2p+h pth 2p+h p+h fên-5p+S6n-6h fên –4p+f6n-5h i=0 n-2 Sei+69+f6i+5k Soi+s9+f6i+4k п-2 foi+3p+ f6i+2h) foi+ap + fei+3h foi+3p + foi+2h foi+69+ fei+sk + i=0 = q foi+59 + f6i+ak, 1+ fon-sp+Sên-6h Sên-4p+ fén-sh i=0 n-2 feit4p+S6i+3h Jei+59+Sei+ak = qT(föi+4P + foi+3h fei+3P + föi+2h n-2 foi+69 + f6i+5k` fei+59 + fei+ak i=0 fon-4p+fon-sh+fén-5p+fën-6h Sên-4P+Sên-sh i=0 п-2 Soi+69+S6i+5k Sei+59+Sei+4k п-2 fei+3p+f6i+2h fei+4P+ fei+3h\ ( foi+69 + fei+5k + i=0 = q|| foi+3P + fei+2h) foi+59 + f6i+ak, fên-3p+ fen–4h Sen-4p+ fen–sh i=0 n-2 (foi+4p+ fei+3h f6i+3p + fsi+2h , foi+6q+ fei+sk foi+5q + fei+ak (fön-4p + fön-sh fön-3p + fön-4h i=0 14
п-2 q II ´fei+4P + fsi+3h\ ( föi+69+ f6i+5k` föi+3P + f6i+2h föi+59 + f6i+4k, i=0 n-2 2p+h p+h f6i+4p+f6i+3h f6i+3P+f6i+2h) f6i+69+f6i+5k Soi+59+S6i+4k i=0 п-2 п-2 п-2 2p+h pth ) 11(Fsi+7p+f6i+6h f6i+2p+f6i+1h II( Foi+1P+f6ih f6i+8p+f6i+7h П f6i+7p+f6i+6h foi+8p+f6i+7h i=0 i=0 i=0 п-2 2p+h p+h q II f6i+4P+f6i+3h f6i+3P+f6i+2h| Sei+59+S6i+4k, п-2 foi+4P + fói+3h` П foi+3P + f6i+2h, ´foi+69 + föi+5k\ I \ foi+39 + föi+ak ) i=0 = q|| n-2 2p+h + f6i+2p+f6i+1h i=0 p+h f6i+1p+f6ih i=0 п-2 2p+h p+h II f6i+4p+f6i+3h f6i+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k foi+69 + fei+5k` ( fei+4P+ foi+3" ) 54 + feitak п-2 i=0 foi+3P + fei+2h, () + ( ( 2p+h p+h 2p+h p+h fón-5p+f6n-6h fon-4p+fén-5h i=0 п-2 П f6i+4P+f6i+3h f6i+3p+f6i+2h) f6i+69+f6i+5k f6i+59+f6i+4k, п-2 (fei+4p+ fei+3h` q II fsi+3p + fei+2h ) (föi+69 + fei+5k + i=0 föi+59 + fói+4k, fon-5p+fon-6h 1+ fon-4p+f6n-5h i=0 п-2 foi+3P+f6i+2h foit69+ Seitsk f6i+59+f6i+4k п-2 II foi+4P + f6i+3h` foi+3P + f6i+2h, ( föi+69 + f6i+5k` + i=0 = q foi+59 + fói+4k, fön-4p+fên-5h+fên-5P+f6n-6h fon-4P+f6n-5h i=0 п-2 f6i+4p+f6i+3h fei+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k п-2 II foi+4P + fei+3h\ fsi+3P + fei+2h ) ( fei+69+ fei+sk` + i=0 foi+59 + fói+4k, fon-3p+f6n-4h fön-4p+fön-5h i=0 п-2 ( föi+4P+ f6i+3h` föi+3P + föi+2h, foi+69 + fei+5k` fön-4P+ fön-šh\1 = q + \ föi+59 + foi+ak ) | fön-3p + fön-4h, i=0 14
Transcribed Image Text:п-2 q II ´fei+4P + fsi+3h\ ( föi+69+ f6i+5k` föi+3P + f6i+2h föi+59 + f6i+4k, i=0 n-2 2p+h p+h f6i+4p+f6i+3h f6i+3P+f6i+2h) f6i+69+f6i+5k Soi+59+S6i+4k i=0 п-2 п-2 п-2 2p+h pth ) 11(Fsi+7p+f6i+6h f6i+2p+f6i+1h II( Foi+1P+f6ih f6i+8p+f6i+7h П f6i+7p+f6i+6h foi+8p+f6i+7h i=0 i=0 i=0 п-2 2p+h p+h q II f6i+4P+f6i+3h f6i+3P+f6i+2h| Sei+59+S6i+4k, п-2 foi+4P + fói+3h` П foi+3P + f6i+2h, ´foi+69 + föi+5k\ I \ foi+39 + föi+ak ) i=0 = q|| n-2 2p+h + f6i+2p+f6i+1h i=0 p+h f6i+1p+f6ih i=0 п-2 2p+h p+h II f6i+4p+f6i+3h f6i+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k foi+69 + fei+5k` ( fei+4P+ foi+3" ) 54 + feitak п-2 i=0 foi+3P + fei+2h, () + ( ( 2p+h p+h 2p+h p+h fón-5p+f6n-6h fon-4p+fén-5h i=0 п-2 П f6i+4P+f6i+3h f6i+3p+f6i+2h) f6i+69+f6i+5k f6i+59+f6i+4k, п-2 (fei+4p+ fei+3h` q II fsi+3p + fei+2h ) (föi+69 + fei+5k + i=0 föi+59 + fói+4k, fon-5p+fon-6h 1+ fon-4p+f6n-5h i=0 п-2 foi+3P+f6i+2h foit69+ Seitsk f6i+59+f6i+4k п-2 II foi+4P + f6i+3h` foi+3P + f6i+2h, ( föi+69 + f6i+5k` + i=0 = q foi+59 + fói+4k, fön-4p+fên-5h+fên-5P+f6n-6h fon-4P+f6n-5h i=0 п-2 f6i+4p+f6i+3h fei+3p+f6i+2h f6i+6q+f6i+5k f6i+59+f6i+4k п-2 II foi+4P + fei+3h\ fsi+3P + fei+2h ) ( fei+69+ fei+sk` + i=0 foi+59 + fói+4k, fon-3p+f6n-4h fön-4p+fön-5h i=0 п-2 ( föi+4P+ f6i+3h` föi+3P + föi+2h, foi+69 + fei+5k` fön-4P+ fön-šh\1 = q + \ föi+59 + foi+ak ) | fön-3p + fön-4h, i=0 14
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