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n-2 = PII6i+6p+f6i+5h\ foi+5p+f6i+4h, C i=0 n-2 (f6i+6p+f6i+5h) Р ›II foi+5P+f6i+4h, i=0 n-2 (f6i+6p+f6i+5h) PII foi+5P+f6i+4h, i=0 Therefore foi+29+f6i+1k f6i+19+ feik n-2 f6i+6p+f6i+5h) foi+29+f6i+1k PII f6i+5p+f6i+4h, foi+19+ feik i=0 i=0 + (foi+29+ foi+1k) f6i+19+ feik ) [₁ n-2 PII i=0 n-1 T6n-4 = h6iP + f6i-1h foi-1P+ foi-2h, 1+ f6i+6p+f6i+sh f6i+29+f6i+1k f6i+5p+f6i+4h foi+29+f6i+1k\ [fon-59 + fen-6k+fon-69 + fon-7k foi+19+ foik + Jen-7k] fon-59+fen-6k fon-59+fen-sk fen-69+fen-7k fon-69+fen-7k fon-59+fen-sk fon-19+fon-5k] fon-59+fen-6k f6i+29+ foi+1k h) (fi+29

n-2 = PII6i+6p+f6i+5h\ foi+5p+f6i+4h, C i=0 n-2 (f6i+6p+f6i+5h) Р ›II foi+5P+f6i+4h, i=0 n-2 (f6i+6p+f6i+5h) PII foi+5P+f6i+4h, i=0 Therefore foi+29+f6i+1k f6i+19+ feik n-2 f6i+6p+f6i+5h) foi+29+f6i+1k PII f6i+5p+f6i+4h, foi+19+ feik i=0 i=0 + (foi+29+ foi+1k) f6i+19+ feik ) [₁ n-2 PII i=0 n-1 T6n-4 = h6iP + f6i-1h foi-1P+ foi-2h, 1+ f6i+6p+f6i+sh f6i+29+f6i+1k f6i+5p+f6i+4h foi+29+f6i+1k\ [fon-59 + fen-6k+fon-69 + fon-7k foi+19+ foik + Jen-7k] fon-59+fen-6k fon-59+fen-sk fen-69+fen-7k fon-69+fen-7k fon-59+fen-sk fon-19+fon-5k] fon-59+fen-6k f6i+29+ foi+1k h) (fi+29

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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X'ôn-4 = hI( foiP+ f6i-1h Toi+19 + foik п-2 f6i+29+S6i+1k f6i+19+f6ik f6i+5P+f6i+4h ) i=0 п-2 = p]I foi+6P + fei+5h` foi+5P + föi+4h, fei+29 + foi+1k fei+19 + feik fön-79+fön-8k fén-69+f6n-7k i=0 1+ п-2 f6i+29+f6i+1k f6i+19+f6ik p (föi+29 + fei+1k` foi+19 + feik п-2 (foi+6P + fei+5h` f6i+5p+f6i+4h ) i=0 foi+5P + föi+4h, fón-69+f6n-7k+fön-79+f6n-8k fön-69+ fen-7k i=0 п-2 PII f6i+6p+f6i+5h fei+5p+f6i+4h) ) ( foi+29+fei+1k f6i+19+f6ik п-2 foi+6P + fei+5h` (föi+29 + f6i+1k` + i=0 II foi+5P+ foi+4h, fei+19 + feik fên-59+f6n-6k fon-69+fön-7k i=0 ( fei+6P + foi+3" ) ( g+ feik п-2 PII foi+5P+ fei+4h / (foi+29 + fei+1k fon-69 + fön-7k] + fön-59 + fön-6k ] i=0 PTT( föi+6P+ foi+sh) furn t f) ( 6+24 + Joi+1k)| fön-59 + fön-6k + fön-69 + fen-zk] п-2 foi+19 + f6ik fön-59 + fön-6k i=0 п-2 feit6P + fci+3h ) ( t fek ) Fon–54 + fon-6k, foi+5P + föi+ah , (fei+29 + fei+1k\ [fon-4q + fön-sk] П i=0 Therefore п-1 foi+29 + f6i+1k\ f6i-1P+ féi-2h, i=0 Also, from Eq.(8), we see that X6n-5X6n-6 X6n-3 = X6n-6+ X6n-5 + x6n-8 п-2 föi+4P + foi+3h` 9II foi+3P + fei+2h) foi+39 + foi+ak, foi+69 + f6i+5k° i=0 n-2 n-2 f6i+4P+f6i+3h ( f6i+8P+f6i+7h` J6i+7P+f6i+6h, f6i+69+f6i+5k` 2p+h p+h J6i+39+f6i+2k i=0 i=0 + п-2 п-2 r( 2p+h pth ( $6i+2P+f6i+1h )( f6i+49+f6i+3k ) +r]I(1P+f6;h )f6i+39+f6i+2k, f6i+8p+f6i+7h\( f6i+49+f6i+3k i=0 i=0 13
Transcribed Image Text:X'ôn-4 = hI( foiP+ f6i-1h Toi+19 + foik п-2 f6i+29+S6i+1k f6i+19+f6ik f6i+5P+f6i+4h ) i=0 п-2 = p]I foi+6P + fei+5h` foi+5P + föi+4h, fei+29 + foi+1k fei+19 + feik fön-79+fön-8k fén-69+f6n-7k i=0 1+ п-2 f6i+29+f6i+1k f6i+19+f6ik p (föi+29 + fei+1k` foi+19 + feik п-2 (foi+6P + fei+5h` f6i+5p+f6i+4h ) i=0 foi+5P + föi+4h, fón-69+f6n-7k+fön-79+f6n-8k fön-69+ fen-7k i=0 п-2 PII f6i+6p+f6i+5h fei+5p+f6i+4h) ) ( foi+29+fei+1k f6i+19+f6ik п-2 foi+6P + fei+5h` (föi+29 + f6i+1k` + i=0 II foi+5P+ foi+4h, fei+19 + feik fên-59+f6n-6k fon-69+fön-7k i=0 ( fei+6P + foi+3" ) ( g+ feik п-2 PII foi+5P+ fei+4h / (foi+29 + fei+1k fon-69 + fön-7k] + fön-59 + fön-6k ] i=0 PTT( föi+6P+ foi+sh) furn t f) ( 6+24 + Joi+1k)| fön-59 + fön-6k + fön-69 + fen-zk] п-2 foi+19 + f6ik fön-59 + fön-6k i=0 п-2 feit6P + fci+3h ) ( t fek ) Fon–54 + fon-6k, foi+5P + föi+ah , (fei+29 + fei+1k\ [fon-4q + fön-sk] П i=0 Therefore п-1 foi+29 + f6i+1k\ f6i-1P+ féi-2h, i=0 Also, from Eq.(8), we see that X6n-5X6n-6 X6n-3 = X6n-6+ X6n-5 + x6n-8 п-2 föi+4P + foi+3h` 9II foi+3P + fei+2h) foi+39 + foi+ak, foi+69 + f6i+5k° i=0 n-2 n-2 f6i+4P+f6i+3h ( f6i+8P+f6i+7h` J6i+7P+f6i+6h, f6i+69+f6i+5k` 2p+h p+h J6i+39+f6i+2k i=0 i=0 + п-2 п-2 r( 2p+h pth ( $6i+2P+f6i+1h )( f6i+49+f6i+3k ) +r]I(1P+f6;h )f6i+39+f6i+2k, f6i+8p+f6i+7h\( f6i+49+f6i+3k i=0 i=0 13
Brn-1n-2 YIn-1 + 8xn-4' = 0, 1, .., In+1 = aIn-2+ (1) The following special case of Eq.(1) has been studied Xn-1In-2 In+1 = In-2 + (8) In-1 + In-4' where the initial conditions r-4, x-3, x-2, x-1,and ro are arbitrary non zero real numbers. Theorem 4. Let {In}-4 be a solution of Eq.(8). Then for n = 0,1, 2, .. п-1 feip + fei-ih ( fei+29 + foi+1k hII fo X6n-4 i-1p+ foi-2h fei+19 + feik i=0 п-1 fei+4p+ fei+3h feig + foi-ik foi+3P + fei+2h) ( Fei-19 + fei-2k) n-1 ( fei+2P + Joi+" a39 + foi+2k , X6n-3 %3D i=0 fei+49+ fei+3k X6n-2 fei+1P + feih i=0 n-1 foi+24+ fei+1k foi+19 + feik foi+6p+ fei+5h PII foi+5p + fei+ah X6n-1 %3D i=0 n-1 П (fei+4p+ foi+3h\ ( foi+69 + föi+5k foi+3p + fei+2h) X6n %3D foi+59 + foi+ak, i=0 п-1 ( föi+8P+ fei+7h П foi+7P + foi+sh foi+49 + f6i+3k fei+39 + f6i+2k) 2р + h T6n+1 %3D p+h i=0 where x-4 = h, x-3 = k, x-2 = r, x-1 = p, xo = q, {fm}m=-1 Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n – 2. That is; 100 {1,0, 1, 1, 2, 3, 5, 8, ...}). п-2 foi+4p + fei+3h fei+3p + fei+2h) ( foi-19 + fei-2k ) foig + föi-ik X6n-9 %3D i=0 п-2 fei+2P + fei+1h( fei+49 + foi+3k I. fei+1p + feih ) X6n-8 %3D fei+39 + fei+2k, i=0 п-2 П ( foi+6P + fei+sh\ ( foi+29 + foi+ik fsi+sP+ fei+ah, X6n-7 %3D fei+14 + foik i=0 п-2 foi+4P + foi+3h föi+3P + fei+2h ) \foi+59 + foi+ak ) foi+69 + foi+sk` qII X6n-6 %3D i=0 n-2 föi+8P + foi+7h foi+7P + fei+sh föi+49 + fei+3k ) Foi+:39 + foi+2k, 2p + h T6n-5 p+h i=0 Now, it follows from Eq.(8) that X6n-6X6n-7 X6n-4 = x6n-7 + X6n-6 + X6n-9 11
Transcribed Image Text:Brn-1n-2 YIn-1 + 8xn-4' = 0, 1, .., In+1 = aIn-2+ (1) The following special case of Eq.(1) has been studied Xn-1In-2 In+1 = In-2 + (8) In-1 + In-4' where the initial conditions r-4, x-3, x-2, x-1,and ro are arbitrary non zero real numbers. Theorem 4. Let {In}-4 be a solution of Eq.(8). Then for n = 0,1, 2, .. п-1 feip + fei-ih ( fei+29 + foi+1k hII fo X6n-4 i-1p+ foi-2h fei+19 + feik i=0 п-1 fei+4p+ fei+3h feig + foi-ik foi+3P + fei+2h) ( Fei-19 + fei-2k) n-1 ( fei+2P + Joi+" a39 + foi+2k , X6n-3 %3D i=0 fei+49+ fei+3k X6n-2 fei+1P + feih i=0 n-1 foi+24+ fei+1k foi+19 + feik foi+6p+ fei+5h PII foi+5p + fei+ah X6n-1 %3D i=0 n-1 П (fei+4p+ foi+3h\ ( foi+69 + föi+5k foi+3p + fei+2h) X6n %3D foi+59 + foi+ak, i=0 п-1 ( föi+8P+ fei+7h П foi+7P + foi+sh foi+49 + f6i+3k fei+39 + f6i+2k) 2р + h T6n+1 %3D p+h i=0 where x-4 = h, x-3 = k, x-2 = r, x-1 = p, xo = q, {fm}m=-1 Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n – 2. That is; 100 {1,0, 1, 1, 2, 3, 5, 8, ...}). п-2 foi+4p + fei+3h fei+3p + fei+2h) ( foi-19 + fei-2k ) foig + föi-ik X6n-9 %3D i=0 п-2 fei+2P + fei+1h( fei+49 + foi+3k I. fei+1p + feih ) X6n-8 %3D fei+39 + fei+2k, i=0 п-2 П ( foi+6P + fei+sh\ ( foi+29 + foi+ik fsi+sP+ fei+ah, X6n-7 %3D fei+14 + foik i=0 п-2 foi+4P + foi+3h föi+3P + fei+2h ) \foi+59 + foi+ak ) foi+69 + foi+sk` qII X6n-6 %3D i=0 n-2 föi+8P + foi+7h foi+7P + fei+sh föi+49 + fei+3k ) Foi+:39 + foi+2k, 2p + h T6n-5 p+h i=0 Now, it follows from Eq.(8) that X6n-6X6n-7 X6n-4 = x6n-7 + X6n-6 + X6n-9 11
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