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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 Un+1 = Qun-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 u; for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs.

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 Un+1 = Qun-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 u; for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs.

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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From Eq. (18), one can have Ugn-7U8n-11 И8п-5 — иsn-7 И8п-9 + usn-11 c2n-1e2n – 1 c2n-2 2n – 2 T?n-1 (ic+a) T?",2(ie+c) II?",2(ic+a) I", c2n-2e2n–2 II",2 (ic+a)I1",(ie+c) c2n-1e2n-1 T2n–3 (ie+c) e2n –1 ç2n–2 ? (ie+c)(iс+а) -2п-1 2n-2 II" (ic + a) II (ie + c) =1 c2n-12n-1 c2n-1e2n- 2 2п-1 2n-2 II (ic + a) II (ie + c) II",' (ic+a) II?",²(ie+c) [I",°(ie+c)(2n-2, -3 i=1 i=1 e I?",3 (ie+e) (ie+c) c2n-1e2n-1 c2n-1e2n-2 -2n-2 II, (ic + a) [I²(ie + c) 2n-1 II (ic + a) II (ie+c) (en-2je+c + 2n-1 i=1 li=1 c2n-1 2n-1 1 1 2п-1 -2п -2 II", (ic + a) II (ie + c) e ((2n–2)e+c + c2n-1e2n-1 e IT", (ic + a) [I,(ie + c) 2n-1 2п-2 (2n — 1)е + с :1 e2n 2n-1 Пе + с) (iс +a) 2n-1 Moreover, Eq. (18) gives us that U8n-6U8n-10 Ugn-4 = U8n-6 И8п-8 + U8n-10 d2n-1 f2n-1 d2n- 2 f2n –2 d2n-1 -1 IT (id+b) I",²(if+d) II?",?(id+b) [I?",³ (if+d) f2n-1 d2n–2 2n-2(if+d)(id+b) T2n-: 2n-3 II"(id + b) IT(if + d) d2n- 2 f2n-2 2n–2 (id+b) [I?"(if+d) 2n-1 =D1 П d2n-1 d2n-1 f2n-2 2n-1 П(а + b) П (if + d) II?",'(id+b) [I?,?(if+d) II?"³(if+d)( Ten-24f+d) =1 =D1 f II", (if+d) d2n-1 f2n-1 d2n-1 f2n-2 IT (id + b) II, (if + d) 2п-1 -2п-2 II (id + b) [I?²(if +d) ( en-2)F+d + d²n-1 f2n–1((2n – 2)f + d) П(id + b) П + d) (2n — 1)/ + d) (2n – 2) f + d\ (2n – 1) f + d ) 2n-1 i=1 i=1 d2n-1 2n-2, li=1 2n-1 II (id + b) IT (if + d) 2п-1 2n- i=1 i=1 d2n-1 1 - II (id + b) [I(if + d) d?n-1 f2n IT (id + b) IT, (if + d)((2n – 1)f + d) f2n ď²n-1 IL", (if + d)(id+ b) 2n-1 =1 i=1 2п-1
Transcribed Image Text:From Eq. (18), one can have Ugn-7U8n-11 И8п-5 — иsn-7 И8п-9 + usn-11 c2n-1e2n – 1 c2n-2 2n – 2 T?n-1 (ic+a) T?",2(ie+c) II?",2(ic+a) I", c2n-2e2n–2 II",2 (ic+a)I1",(ie+c) c2n-1e2n-1 T2n–3 (ie+c) e2n –1 ç2n–2 ? (ie+c)(iс+а) -2п-1 2n-2 II" (ic + a) II (ie + c) =1 c2n-12n-1 c2n-1e2n- 2 2п-1 2n-2 II (ic + a) II (ie + c) II",' (ic+a) II?",²(ie+c) [I",°(ie+c)(2n-2, -3 i=1 i=1 e I?",3 (ie+e) (ie+c) c2n-1e2n-1 c2n-1e2n-2 -2n-2 II, (ic + a) [I²(ie + c) 2n-1 II (ic + a) II (ie+c) (en-2je+c + 2n-1 i=1 li=1 c2n-1 2n-1 1 1 2п-1 -2п -2 II", (ic + a) II (ie + c) e ((2n–2)e+c + c2n-1e2n-1 e IT", (ic + a) [I,(ie + c) 2n-1 2п-2 (2n — 1)е + с :1 e2n 2n-1 Пе + с) (iс +a) 2n-1 Moreover, Eq. (18) gives us that U8n-6U8n-10 Ugn-4 = U8n-6 И8п-8 + U8n-10 d2n-1 f2n-1 d2n- 2 f2n –2 d2n-1 -1 IT (id+b) I",²(if+d) II?",?(id+b) [I?",³ (if+d) f2n-1 d2n–2 2n-2(if+d)(id+b) T2n-: 2n-3 II"(id + b) IT(if + d) d2n- 2 f2n-2 2n–2 (id+b) [I?"(if+d) 2n-1 =D1 П d2n-1 d2n-1 f2n-2 2n-1 П(а + b) П (if + d) II?",'(id+b) [I?,?(if+d) II?"³(if+d)( Ten-24f+d) =1 =D1 f II", (if+d) d2n-1 f2n-1 d2n-1 f2n-2 IT (id + b) II, (if + d) 2п-1 -2п-2 II (id + b) [I?²(if +d) ( en-2)F+d + d²n-1 f2n–1((2n – 2)f + d) П(id + b) П + d) (2n — 1)/ + d) (2n – 2) f + d\ (2n – 1) f + d ) 2n-1 i=1 i=1 d2n-1 2n-2, li=1 2n-1 II (id + b) IT (if + d) 2п-1 2n- i=1 i=1 d2n-1 1 - II (id + b) [I(if + d) d?n-1 f2n IT (id + b) IT, (if + d)((2n – 1)f + d) f2n ď²n-1 IL", (if + d)(id+ b) 2n-1 =1 i=1 2п-1
This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 dun-5 Un+1 = aun-1+ п 3D0, 1,..., (1) Yun-3 Bun-1un-5 Yun-3 + dun-5 Un+1 = Qun-1 n = 0,1, .., (2) 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 = 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 11. EXACT SOLUTION OF EQ. (2) WHEN a = B =y= 6 = 1 This section shows the exact solutions of the following equation: Un-1Un-5 Un+1 = Un-1 n = 0, 1,..., (18) Un-3 + Un-5 where the initial conditions are selected to be positive real numbers. Theorem 9 Let {un}-5 be a solution to Eq. (18) and suppose that u-5 а, и-4 %3D 6, и-з 3D с, и-2 %3D d, и-1 %3D е, ио 3D f. Then, for n %3D 0, 1,2, ..., the solutions of Eq. (18) are given by the following formulas: %3D e2n 2n-1 II (ie +c) (ic+a)' f2n d2n-1 IIT (if+ d)(id + b)' c2n e2n U8n-5 2n-1 i=1 U8n-4 = -2n-1 U8n-3 = 2n- IT (ic + a) I (ie + c) d2n f2n U8n-2 = II (id + b) IIT (if + d)' e2n+1 2n i=1 U8n-1 = II, (ie + c)(ic + a)' U8n = IT (if + d)(id + b)' c2n+1e2n+1 U8n+1 = 72n+1 2n IT (ic + a) I, (ie + c) d2n+1 f2n+1 IT (id + b) II(if + d)" =D1 U8n+2 = n+1 Proof. It can be easily seen that the solutions are true for n = 0. We suppose that n >0 and assume that our assumption holds for n- 1. That is, e2n-2,2n-3 2n-3 U8n-13 = II (ie + c) (ic + a) f2n-2 d2n-3 2n-3 II" (if + d)(id + b) c2n-22n-2 U8n-12 = U8n-11 = 2n-2 II (ic + a) I (ie + c) d2n-2 f2n-2 II (id + b) IT (if + d) e2n-12n-2 U8n-10 = 2n-3 li=1 2n-2 U8n-9 = T2n-2 I (ie + c)(ic + a) f2n-12n-2 IT(if + d)(id+b) U8n-8 li=1 can-1e2n-1 U8n-7 = -2n-1 2n-2 :1 II (ic+ a) II (ie+c) d2n-1 f2n-1 U8n-6 = 2n-1 II (id + b) II (if + d)
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 dun-5 Un+1 = aun-1+ п 3D0, 1,..., (1) Yun-3 Bun-1un-5 Yun-3 + dun-5 Un+1 = Qun-1 n = 0,1, .., (2) 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 = 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 11. EXACT SOLUTION OF EQ. (2) WHEN a = B =y= 6 = 1 This section shows the exact solutions of the following equation: Un-1Un-5 Un+1 = Un-1 n = 0, 1,..., (18) Un-3 + Un-5 where the initial conditions are selected to be positive real numbers. Theorem 9 Let {un}-5 be a solution to Eq. (18) and suppose that u-5 а, и-4 %3D 6, и-з 3D с, и-2 %3D d, и-1 %3D е, ио 3D f. Then, for n %3D 0, 1,2, ..., the solutions of Eq. (18) are given by the following formulas: %3D e2n 2n-1 II (ie +c) (ic+a)' f2n d2n-1 IIT (if+ d)(id + b)' c2n e2n U8n-5 2n-1 i=1 U8n-4 = -2n-1 U8n-3 = 2n- IT (ic + a) I (ie + c) d2n f2n U8n-2 = II (id + b) IIT (if + d)' e2n+1 2n i=1 U8n-1 = II, (ie + c)(ic + a)' U8n = IT (if + d)(id + b)' c2n+1e2n+1 U8n+1 = 72n+1 2n IT (ic + a) I, (ie + c) d2n+1 f2n+1 IT (id + b) II(if + d)" =D1 U8n+2 = n+1 Proof. It can be easily seen that the solutions are true for n = 0. We suppose that n >0 and assume that our assumption holds for n- 1. That is, e2n-2,2n-3 2n-3 U8n-13 = II (ie + c) (ic + a) f2n-2 d2n-3 2n-3 II" (if + d)(id + b) c2n-22n-2 U8n-12 = U8n-11 = 2n-2 II (ic + a) I (ie + c) d2n-2 f2n-2 II (id + b) IT (if + d) e2n-12n-2 U8n-10 = 2n-3 li=1 2n-2 U8n-9 = T2n-2 I (ie + c)(ic + a) f2n-12n-2 IT(if + d)(id+b) U8n-8 li=1 can-1e2n-1 U8n-7 = -2n-1 2n-2 :1 II (ic+ a) II (ie+c) d2n-1 f2n-1 U8n-6 = 2n-1 II (id + b) II (if + d)
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