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-5dun-5Un+1 = aun-1+п 3D0, 1,...,(1)Yun-3Bun-1un-5Yun-3 + dun-5Un+1 = Qun-1n = 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 alsopresent the numerical solutions via some 2D graphs.2. ON THE EQUATION Un+1 = QUn-1+Bun-1un-5yun-3-dun-5This section is devoted to study the qualitative behaviors of Eq. (1). Theequilibrium point of Eq. (1) is given by11. EXACT SOLUTION OF EQ. (2) WHEN a = B =y= 6 = 1This section shows the exact solutions of the following equation:Un-1Un-5Un+1 = Un-1n = 0, 1,...,(18)Un-3 + Un-5where 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-5а, 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, ..., thesolutions of Eq. (18) are given by the following formulas:%3De2n 2n-1II (ie +c) (ic+a)'f2n d2n-1IIT (if+ d)(id + b)'c2n e2nU8n-52n-1i=1U8n-4 =-2n-1U8n-3 =2n-IT (ic + a) I (ie + c)d2n f2nU8n-2 =II (id + b) IIT (if + d)'e2n+1 2ni=1U8n-1 =II, (ie + c)(ic + a)'U8n =IT (if + d)(id + b)'c2n+1e2n+1U8n+1 =72n+12nIT (ic + a) I, (ie + c)d2n+1 f2n+1IT (id + b) II(if + d)"=D1U8n+2 =n+1Proof.It can be easily seen that the solutions are true for n = 0. We supposethat n >0 and assume that our assumption holds for n- 1. That is,e2n-2,2n-32n-3U8n-13 =II (ie + c) (ic + a)f2n-2 d2n-32n-3II" (if + d)(id + b)c2n-22n-2U8n-12 =U8n-11 =2n-2II (ic + a) I (ie + c)d2n-2 f2n-2II (id + b) IT (if + d)e2n-12n-2U8n-10 =2n-3li=12n-2U8n-9 =T2n-2I (ie + c)(ic + a)f2n-12n-2IT(if + d)(id+b)U8n-8li=1can-1e2n-1U8n-7 =-2n-12n-2:1II (ic+ a) II (ie+c)d2n-1 f2n-1U8n-6 =2n-1II (id + b) II (if + d) In addition, Eq. (18) leads toU8n-5U8n-9U8n-3 = U8n-5И8п-7 + u8n-9e2n 2n –1e2n-1,2n-2e2n c2n-1(ie+c)(ic+a) [I",?(ie+c)(ic+a)e2n-1c2n-2IT",?(ie+c)(ic+a)c2n-2e2nc2n-1e2n-1II?",(ic+a) [I?",²(ie+c)2n-1П (е + c) (iс + а)i=D11.e2n c2n-1II ie + e)(ic + a) II, "(ie + c)(ic +a) II (ic + a) (TT Geta) + eIlie-2n-11li=1II (ic+a)i=1cII"?(ic+a)e2n 2n-1c2n-22nIT (ie + c)(ic +a) II'(ie +c)(ic + a) ( en-ie+a +)2n-1li=12n-1c2n-1e2n ((2n – 1)c+a)ПТ (iе + с) (іс +а) ПТ (ie + с) (iс + а) (2m)с + а)(2n — 1)с + а(2n)с + аe2n c2n-12n-1li=12п-1li=1e2n c2n-1(1II", (ie + c)(ic +a)2n-1li=1e2n 2nc2n e2n2n-1Li=12n-1-2nП (iе + с) (ic+ta)((2n)с + а)П (с + а) П-T (ie + c)i=1i=1Similarly, one can prove other solutions.

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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 d are positive real numbers and the initial con- ditions ui for all i = present the numerical solutions via some 2D graphs. -5, -4, ..., 0, are arbitrary non-zero real numbers. We also

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 d are positive real numbers and the initial con- ditions ui for all i = present the numerical solutions via some 2D graphs. -5, -4, ..., 0, are arbitrary non-zero real numbers. We also

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.7: Problem Solving: Consecutive Integers

Problem 14OE

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