Show me the steps of determine green and all information is here This work aims to investigate the equilibria, local stability, global attractivityand the exact solutions of the following difference equationsBun-1un-5Un+1 = aUn-1+n = 0,1, ..,(1)YUn-3 - dun-5'Bun-1un-5Un+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 alsopresent the numerical solutions via some 2D graphs.2. ON THE EQUATION Un+1 = aUn-1 +Bun-1un-5Yun-3-dun-5(1). TheThis section is devoted to study the qualitative behaviors of Eq.equilibrium point of Eq. (1) is given by6. EXACT SOLUTION OF EQ. (1) WHEN a =B=y= 8 = 1In this section, we investigate the exact solutions of the following rational differ-ence equationUn-1un-5Иn+1 — ит-1 +n = 0, 1, ...,(10)Un-3 - Un-5where the initial conditions are positive real numbers.Theorem 5 Let {un}n=-5be a solution to Eq. (10) and suppose that u-5 =d, u-1 = e, uo = f. Then, for n = 0, 1,2, .., theа, 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+le2nn-1U8n-4 =U8n-3 =a" (a – c)" (c – e)"'dn+1 f2nb" (b – d)" (d- f)"'e2n+1cnU8n-2 =U8n-1 =a" (a – c)"(c – e)* 'f2n+1gnbr (b – d)"(d – f)"'cn+1 e2n+1U8nU8n+1a" (c - e)" (a – c)n+1>dn+1 f2n+1b" (d – f)"(b – d)n+1*U8n+2 =Proof.It can be easily observed that the solutions are true for n = 0. We nowassume that n > 0 and that our assumption holds for n – 1. That is,e2n-2 cn-1n-2(c – e)n-1(a – c)n-1'f2n-2 an-1b2-2 (d – f)n-1(b – d)a-I?U8n-13 =U8n-12 =c"e2n-2U8n-11n-1(a - c)a-1(c – e)n–1'an-1 d" f2n-2(b – d)"-1(d – f)"-1’e2n-1 cn-1И8п-10bn-1U8n-91(a – c)n-1(c – e)"-I’f2n-1qn-1(b – d)n-1(d – f)n-I³an-U8n-8 =bn-1c"e2n-1U8n-7 =n-1(c – e)n-1(a – c)n'd" f2n-1br-1 (d – f)n-1(b – d)nan-U8n-6 =Eq. (10) gives us thatU8n-7u8n-11И8п-5 — и8n-7 +и8п-9 — и8n-11cne2n-1c"e2n-1c"e2n-2-1 (с — е)"-1 (а — с)"an-1(c-e) n-1(a-c)n an-1(a-c)n-1(c-e)n –1e2n–1 cn-1c)n-1anспе2n -2an-(с-е)п-Т(а-с)п-an-I(c-e)n-Ic"e2n-1e2n-2 n(c – e)n–1an-- c)"1(e – e)n-1(a – c)" ( - )а —an-1c"e2n-1e2n-1 n+1+– e)n-1(ae2n cn— с)"an-1(c- e)"(a – c)"an-1с —-'(c – e)" (a – c)™ *an-1Moreover, it can be seen from Eq. (10) thatU8n-6U8n-10И8п—4 — и8n-6 +И8п-8 — и8n-10d" f2n-1d" f2n-1br-1(d – f)"-1(b – d)"dn f2n-2bn-1(d-f)n-1(b-d)n bn-1(b-d)n-1 (d-f)n-If2n-1dn-1bn-1(b-d)n -1(d-f)n-1dn f2n-2bn-1(b-d) n(d-f)n-1d" f2n-1br-1 (d — f)"-1(ь — d)"dr f2n-2bn-1 (d – f)"-1(b – d)" (à – })d" f2n-1bn-1(d – f)"-1(b – d)nf2m drbn-1(d – f)"(b – d)"°dn+1 f2n-1bn-1 (d – f)" (b – d)"

Given equation is as follows: un+1=un-1+un-1un-5un-3-un-5, n=0, 1, ...............(10) The solution…

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 = Qun-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 u; 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 Uln+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 Un+1 = Qun-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 u; 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 Uln+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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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