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(i) If A 0 and B‡0, then 1-k₁+ko 1-ko-k₁ fly C (1) lim W2n →∞ On the other hand, if wo = w_2 = and W2n = wo = w_2. (ii) If A + 0, B #0 and W2n-1= ot√²-4ko k1 2 ko lim w2n-1= 818 Otherwise, if wo = w-2 = (iii) If C 0 and D = 0, then ∞, 818 1-ki+ko 1-ko-ki lim 22 n () ((1 — ko) wo − μ) = w-1. - - 1-k1+ko 1-ko-k₁ #1, then 1-ko+k1 ((to − 1) (=ko+ko) - ko) μ₂ 1-k1-ko and and 1-v1+v0 1-10-21 |(*= √6²_-4ko k1)| 2 1) ₁₂₁ | ( €, ot√2-4 ko k1 2 662-4ko k1 2 o+√²-4ko k1 2 6±√62-4ko k1 2 6±√²-4ko k1 2 √√²-410 v1 2 √√√²-4 vo v1 2 < 1, > 1. < 1, > 1. 0, then A= B =0 < 1, 0, then A= B = 0 and > 1.

(i) If A 0 and B‡0, then 1-k₁+ko 1-ko-k₁ fly C (1) lim W2n →∞ On the other hand, if wo = w_2 = and W2n = wo = w_2. (ii) If A + 0, B #0 and W2n-1= ot√²-4ko k1 2 ko lim w2n-1= 818 Otherwise, if wo = w-2 = (iii) If C 0 and D = 0, then ∞, 818 1-ki+ko 1-ko-ki lim 22 n () ((1 — ko) wo − μ) = w-1. - - 1-k1+ko 1-ko-k₁ #1, then 1-ko+k1 ((to − 1) (=ko+ko) - ko) μ₂ 1-k1-ko and and 1-v1+v0 1-10-21 |(*= √6²_-4ko k1)| 2 1) ₁₂₁ | ( €, ot√2-4 ko k1 2 662-4ko k1 2 o+√²-4ko k1 2 6±√62-4ko k1 2 6±√²-4ko k1 2 √√²-410 v1 2 √√√²-4 vo v1 2 < 1, > 1. < 1, > 1. 0, then A= B =0 < 1, 0, then A= B = 0 and > 1.

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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5 Asymptotic behavior of the solution This section discusses the behavior and boundedness of the solutions of (4). Theorem 5.1. Suppose that {wn, zn}_, is a solution of (4), then the following statements n=-2 are holds. 10 (i) If A + 0 and B + 0, then |(* $±Vø2-4 ko k1 1-kj+ko 1-ko-ki ) H, < 1, lim w2nl n 00 0tV$2-4 ko k1 o, > 1. 2 p±/$2-4 ko kı On the other hand, if wo = w_2 = 1-kı+ko 1-ko-ki + 0, then A = B = 0 H and 2 and w2n = wo = w_2· (ii) If A # 0, B+ 0 and 0±/02-4 ko kı 2 ko # 1, then | ((6-1) (는부북)-) 시 ( $±V¢2-4 ko k1 1-ko+k1 1-k1-ko < 1, ko lim w2 n-1| = n 00 $±V¢2-4 ko k1 > 1. +V62-4 ko kı Otherwise, if wo = w_2 = 1-k1+ko 1-ko-ki u and + 0, then A = B = 0 and 2 ) ((1– ko) wo – H) = W2n-1 = ko = W-1. (iii) If C + 0 and D+ 0, then t/2-4 vo v1 1-vi+vo 1-v0-v1 < 1, E, lim 2n| = n 00 t/2 -4 vo v1 0, > 1. 2 2–4 vo v1 On the other hand, if zo = z-2 = 1-vi+vo 1-vo-vi e and + 0, then C = D = 0 and z2n = 20 = 2–2. tV2-4 vo v1 (iv) If C # 0, D # 0 and # 1, then 2 vo ((- 1) () - ) (Viorn tV2-4 vovị 1-vo+vi 1-v1-vo < 1, E, lim 2n-1| = n 00 b£/2 -4 vo v1 > 1. t/2-4 vo v1 Otherwise, if zo = z_2 = 1-vi+v0 1-v0-v1 € and + 0, then C = D = 0 and (5) ((1 – vo) žo – e) 22n-1 = = 2-1. VO 11
Transcribed Image Text:5 Asymptotic behavior of the solution This section discusses the behavior and boundedness of the solutions of (4). Theorem 5.1. Suppose that {wn, zn}_, is a solution of (4), then the following statements n=-2 are holds. 10 (i) If A + 0 and B + 0, then |(* $±Vø2-4 ko k1 1-kj+ko 1-ko-ki ) H, < 1, lim w2nl n 00 0tV$2-4 ko k1 o, > 1. 2 p±/$2-4 ko kı On the other hand, if wo = w_2 = 1-kı+ko 1-ko-ki + 0, then A = B = 0 H and 2 and w2n = wo = w_2· (ii) If A # 0, B+ 0 and 0±/02-4 ko kı 2 ko # 1, then | ((6-1) (는부북)-) 시 ( $±V¢2-4 ko k1 1-ko+k1 1-k1-ko < 1, ko lim w2 n-1| = n 00 $±V¢2-4 ko k1 > 1. +V62-4 ko kı Otherwise, if wo = w_2 = 1-k1+ko 1-ko-ki u and + 0, then A = B = 0 and 2 ) ((1– ko) wo – H) = W2n-1 = ko = W-1. (iii) If C + 0 and D+ 0, then t/2-4 vo v1 1-vi+vo 1-v0-v1 < 1, E, lim 2n| = n 00 t/2 -4 vo v1 0, > 1. 2 2–4 vo v1 On the other hand, if zo = z-2 = 1-vi+vo 1-vo-vi e and + 0, then C = D = 0 and z2n = 20 = 2–2. tV2-4 vo v1 (iv) If C # 0, D # 0 and # 1, then 2 vo ((- 1) () - ) (Viorn tV2-4 vovị 1-vo+vi 1-v1-vo < 1, E, lim 2n-1| = n 00 b£/2 -4 vo v1 > 1. t/2-4 vo v1 Otherwise, if zo = z_2 = 1-vi+v0 1-v0-v1 € and + 0, then C = D = 0 and (5) ((1 – vo) žo – e) 22n-1 = = 2-1. VO 11
In this paper, we solve and study the properties of the following system E Wn-p > Zn-h Zn-h Wn-P p=0 h=1 p=1 h=0 Wn+1 +µ and zn+1 + €, (4) Zn - € Wn - H where u and e are arbitrary positive real numbers with initial conditions w; and z; for i = -2, –1,0. Theorem 2.1. Let {wn, zn}- be a solution of (4), then n=-2 o+ V02 – 4 ko k1 $ - V62 – 4 ko k1 k1 + ko w2n-2 + B + 2 2 ko – k1 n $ + Vø2 – 4 ko k1 $+ Vø2 – 4 ko k1 0 - V02 – 4 ko k1 0 - Vo2 – 4 ko k1 1 A ko 1 + B ko w2n-1 - 1 - 1 2 2 + (-) (는)-) (1– ko + k1 1– k1 – ko 1 - 1 ko + V2 – 4 vo vi 1 – vị + vo + - V62 4 vo v1 22n-2 + D E, 2 2 - vo - v1 1 + V2 – 4 vọ v1 b + V2 – 4 vo v1 e;2 – 4 vo v1 al2 - 4 vo vị 22n-1 C + D - 1 2 2 2 +(G-) )-) . 1- vi + vo 1- vo - v1 E, where A, B, C and D are constants defined as wo - u z-1 + z-2 ko ki = O = ko + k1 + ko k1, w-1+ w-2 20 - E 20 - € w-1 + w-2 v1 = = vo + vi + vo v1, 2-1+ z-2 1 – Om Vo2 – 4 ko k1 2 62 – 8 ko ki 1– k1 + ko 1 – 1 – kị + ko` 1 — ко — k1 A = - 4 ko k1 – o) + 2 ( wo – w-2 - V62 – 4 ko ki 2 ф2 — 8 kо k1 1 – ko + k1 - ko – k1 (1 – kị + ko) µ)|, 1 – k1 – ko B u - w-2 62 – 4 ko k1 + ¢) +2 ( wo – = 4 vo v1 2 2 – 8 vo v1 1- vi + vo 1- vị + vo 2-2 - 1- vo - vi 1- vo - vi V2 – 4 vo vi 1 – vị + vo 1- v1 + vo E - z-2 2 – 4 vo v1 + +2 20 - 2 2 – 8 vo vi - vO - v1 1- vo - v1 since ko + ki 1 and vo +v1 71 for n € N. Proof. To obtain the expressions of the general solutions for (4), we rewrite it in the follow form Zn-h Wn-p Wn+1 - H h=1 Zn+1 - € and (5) 1 1 Zn - € Wn - u E Wn-p > Zn-h p=0 h=0 4 || ||
Transcribed Image Text:In this paper, we solve and study the properties of the following system E Wn-p > Zn-h Zn-h Wn-P p=0 h=1 p=1 h=0 Wn+1 +µ and zn+1 + €, (4) Zn - € Wn - H where u and e are arbitrary positive real numbers with initial conditions w; and z; for i = -2, –1,0. Theorem 2.1. Let {wn, zn}- be a solution of (4), then n=-2 o+ V02 – 4 ko k1 $ - V62 – 4 ko k1 k1 + ko w2n-2 + B + 2 2 ko – k1 n $ + Vø2 – 4 ko k1 $+ Vø2 – 4 ko k1 0 - V02 – 4 ko k1 0 - Vo2 – 4 ko k1 1 A ko 1 + B ko w2n-1 - 1 - 1 2 2 + (-) (는)-) (1– ko + k1 1– k1 – ko 1 - 1 ko + V2 – 4 vo vi 1 – vị + vo + - V62 4 vo v1 22n-2 + D E, 2 2 - vo - v1 1 + V2 – 4 vọ v1 b + V2 – 4 vo v1 e;2 – 4 vo v1 al2 - 4 vo vị 22n-1 C + D - 1 2 2 2 +(G-) )-) . 1- vi + vo 1- vo - v1 E, where A, B, C and D are constants defined as wo - u z-1 + z-2 ko ki = O = ko + k1 + ko k1, w-1+ w-2 20 - E 20 - € w-1 + w-2 v1 = = vo + vi + vo v1, 2-1+ z-2 1 – Om Vo2 – 4 ko k1 2 62 – 8 ko ki 1– k1 + ko 1 – 1 – kị + ko` 1 — ко — k1 A = - 4 ko k1 – o) + 2 ( wo – w-2 - V62 – 4 ko ki 2 ф2 — 8 kо k1 1 – ko + k1 - ko – k1 (1 – kị + ko) µ)|, 1 – k1 – ko B u - w-2 62 – 4 ko k1 + ¢) +2 ( wo – = 4 vo v1 2 2 – 8 vo v1 1- vi + vo 1- vị + vo 2-2 - 1- vo - vi 1- vo - vi V2 – 4 vo vi 1 – vị + vo 1- v1 + vo E - z-2 2 – 4 vo v1 + +2 20 - 2 2 – 8 vo vi - vO - v1 1- vo - v1 since ko + ki 1 and vo +v1 71 for n € N. Proof. To obtain the expressions of the general solutions for (4), we rewrite it in the follow form Zn-h Wn-p Wn+1 - H h=1 Zn+1 - € and (5) 1 1 Zn - € Wn - u E Wn-p > Zn-h p=0 h=0 4 || ||
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