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This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 YUn-3 - Sun-5 Un+1 = Qun-1+ n = 0,1,.., (1) Bun-1un-5 Yun-3 + Sun-5 where the coefficients a, B, y, and d are positive real numbers and the initial con- ditions u; for all i = -5, –4,..., 0, are arbitrary non-zero real numbers. We also Un+1 = aUn-1 n = 0,1,., (2) present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Un+1 = QUn-1 + Bun-1un-5 yun-3-dun-5 This section is devoted to study the qualitative behaviors of Eq. (1). The equilibrium point of Eq. (1) is given by

This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 YUn-3 - Sun-5 Un+1 = Qun-1+ n = 0,1,.., (1) Bun-1un-5 Yun-3 + Sun-5 where the coefficients a, B, y, and d are positive real numbers and the initial con- ditions u; for all i = -5, –4,..., 0, are arbitrary non-zero real numbers. We also Un+1 = aUn-1 n = 0,1,., (2) present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Un+1 = QUn-1 + Bun-1un-5 yun-3-dun-5 This section is devoted to study the qualitative behaviors of Eq. (1). The equilibrium point of Eq. (1) is given by

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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This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 Un+1 = aUn-1+ n = 0,1, .., (1) YUn-3 - dun-5' Bun-1un-5 Un+1 = aUn-1- n = 0,1, .., (2) YUn-3 + dun-5' where the coefficients a, B, y, and & are positive real numbers and the initial con- ditions ui for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Un+1 = aUn-1 + Bun-1un-5 Yun-3-dun-5 (1). The This section is devoted to study the qualitative behaviors of Eq. equilibrium point of Eq. (1) is given by 6. EXACT SOLUTION OF EQ. (1) WHEN a = B=y= 8 = 1 In this section, we investigate the exact solutions of the following rational differ- ence equation Un-1un-5 Иn+1 — ит-1 + n = 0, 1, ..., (10) Un-3 - Un-5 where the initial conditions are positive real numbers. Theorem 5 Let {un}n=-5 be a solution to Eq. (10) and suppose that u-5 = d, u-1 = e, uo = f. Then, for n = 0, 1,2, .., the а, и-4 — б, и_з — с, и-2 — solutions of Eq. (10) are given by the following formulas: e2n c" U8n-5 = (c- e)"(a – c)n' f2n d" bn-1(d - f)"(b– d)n' cn+le2n n-1 U8n-4 = U8n-3 = a" (a – c)" (c – e)"' dn+1 f2n b" (b – d)" (d- f)"' e2n+1cn U8n-2 = U8n-1 = a" (a – c)"(c – e)* ' f2n+1gn br (b – d)"(d – f)"' cn+1 e2n+1 U8n U8n+1 a" (c - e)" (a – c)n+1> dn+1 f2n+1 b" (d – f)"(b – d)n+1* U8n+2 = Proof. It can be easily observed that the solutions are true for n = 0. We now assume that n > 0 and that our assumption holds for n – 1. That is, e2n-2 cn-1 n-2(c – e)n-1(a – c)n-1' f2n-2 an-1 b2-2 (d – f)n-1(b – d)a-I? U8n-13 = U8n-12 = c"e2n-2 U8n-11 n-1(a - c)a-1(c – e)n–1' an-1
Transcribed Image Text:This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 Un+1 = aUn-1+ n = 0,1, .., (1) YUn-3 - dun-5' Bun-1un-5 Un+1 = aUn-1- n = 0,1, .., (2) YUn-3 + dun-5' where the coefficients a, B, y, and & are positive real numbers and the initial con- ditions ui for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Un+1 = aUn-1 + Bun-1un-5 Yun-3-dun-5 (1). The This section is devoted to study the qualitative behaviors of Eq. equilibrium point of Eq. (1) is given by 6. EXACT SOLUTION OF EQ. (1) WHEN a = B=y= 8 = 1 In this section, we investigate the exact solutions of the following rational differ- ence equation Un-1un-5 Иn+1 — ит-1 + n = 0, 1, ..., (10) Un-3 - Un-5 where the initial conditions are positive real numbers. Theorem 5 Let {un}n=-5 be a solution to Eq. (10) and suppose that u-5 = d, u-1 = e, uo = f. Then, for n = 0, 1,2, .., the а, и-4 — б, и_з — с, и-2 — solutions of Eq. (10) are given by the following formulas: e2n c" U8n-5 = (c- e)"(a – c)n' f2n d" bn-1(d - f)"(b– d)n' cn+le2n n-1 U8n-4 = U8n-3 = a" (a – c)" (c – e)"' dn+1 f2n b" (b – d)" (d- f)"' e2n+1cn U8n-2 = U8n-1 = a" (a – c)"(c – e)* ' f2n+1gn br (b – d)"(d – f)"' cn+1 e2n+1 U8n U8n+1 a" (c - e)" (a – c)n+1> dn+1 f2n+1 b" (d – f)"(b – d)n+1* U8n+2 = Proof. It can be easily observed that the solutions are true for n = 0. We now assume that n > 0 and that our assumption holds for n – 1. That is, e2n-2 cn-1 n-2(c – e)n-1(a – c)n-1' f2n-2 an-1 b2-2 (d – f)n-1(b – d)a-I? U8n-13 = U8n-12 = c"e2n-2 U8n-11 n-1(a - c)a-1(c – e)n–1' an-1
Furthermore, Eq. (10) gives us U8n-5U8n-9 И8п-3 — U8n-5 + U8n-7 - Ugn-9 e2n c" 2n-1n-1 e2n c" + a"-1(c – e)" (a – c)" ап-1(с-е)п (а-с)п ап-1(а—с)п-1(с-е) -1 e2n-1cn-1 ап-1 (а—с)п—1(с-е)п-1 спе2n-1 an-1(c-e)n-1(a-c)n 2n c" e2n cn a"-1 (с — е)"(а — с)" e2n cn а" (с — е)" (а — с)т-т (음-) а — с (с — е)" (а — с)" e2n cn+1 a" (с — е)"(а — с)" an an Also, Eq. (10) leads to U8n-4u8n-8 U8n-2 = U8n-4 + U8n-6 U8n-8 f2n-1 an-1 fn-1(d-f)r(b-d)n bn-1(b-d)n-1(d-f)n-I f2n–1 dn-1 (b-d)n-1(d-f)n-I f2n dn f2n d" bn-1(d – f)"(b – d)" + dn f2n-1 bn-1(d-f)n–1(b-d)n hn f2n d" b* (d – f)"(b – d)n-1 f2n dn bn-1(d – f)"(b –- d)" f2m d" br (d – f)"(b – d)n f2n qn+1 b" (d – f)"(b – d)" 1 d)-1 Moreover, Eq. (10) gives U8n-3U8n-7 U8n-1 = U8n-3 + и8п-5 — и8n-7 cn+1e2n а" (а-с)" (с—е)" e2n cn ап-1(с-е)" (а-с)" c"2n-1 cn+le2n aп-1(с-е)m — 1(а-с)" а" (а — с)"(с — е)" che2n-1 aп-1(с-е)m —1(а-с)т an cn+1e2n c"e2n a" (а — с)"(с — е)" a" (с — е)"-1(а — с)" (1 + c-e n cn+1e2n c"e2n a" (а — с)" (с — е)" а" (с - е)п-1(а — с)п c" e2n+1 a" (a – c)"(c – e)" Finally, Eq. (10) gives
Transcribed Image Text:Furthermore, Eq. (10) gives us U8n-5U8n-9 И8п-3 — U8n-5 + U8n-7 - Ugn-9 e2n c" 2n-1n-1 e2n c" + a"-1(c – e)" (a – c)" ап-1(с-е)п (а-с)п ап-1(а—с)п-1(с-е) -1 e2n-1cn-1 ап-1 (а—с)п—1(с-е)п-1 спе2n-1 an-1(c-e)n-1(a-c)n 2n c" e2n cn a"-1 (с — е)"(а — с)" e2n cn а" (с — е)" (а — с)т-т (음-) а — с (с — е)" (а — с)" e2n cn+1 a" (с — е)"(а — с)" an an Also, Eq. (10) leads to U8n-4u8n-8 U8n-2 = U8n-4 + U8n-6 U8n-8 f2n-1 an-1 fn-1(d-f)r(b-d)n bn-1(b-d)n-1(d-f)n-I f2n–1 dn-1 (b-d)n-1(d-f)n-I f2n dn f2n d" bn-1(d – f)"(b – d)" + dn f2n-1 bn-1(d-f)n–1(b-d)n hn f2n d" b* (d – f)"(b – d)n-1 f2n dn bn-1(d – f)"(b –- d)" f2m d" br (d – f)"(b – d)n f2n qn+1 b" (d – f)"(b – d)" 1 d)-1 Moreover, Eq. (10) gives U8n-3U8n-7 U8n-1 = U8n-3 + и8п-5 — и8n-7 cn+1e2n а" (а-с)" (с—е)" e2n cn ап-1(с-е)" (а-с)" c"2n-1 cn+le2n aп-1(с-е)m — 1(а-с)" а" (а — с)"(с — е)" che2n-1 aп-1(с-е)m —1(а-с)т an cn+1e2n c"e2n a" (а — с)"(с — е)" a" (с — е)"-1(а — с)" (1 + c-e n cn+1e2n c"e2n a" (а — с)" (с — е)" а" (с - е)п-1(а — с)п c" e2n+1 a" (a – c)"(c – e)" Finally, Eq. (10) gives
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