How to deduce this equation from Equation 1 Explain to me the method. Show me the steps of determine green and equation 1 is second pic 2 The local stability of the solutionsThe equilibrium point x of Eq.(1) is the positive solutionof the equationX= (A+B+C+D)x+(6)(d- e)x'|where d+ e. If [(A+B+C+D) – 1] (e – d) > 0, thenthe only positive equilibrium point ã of Eq.(1) is given byX=[(A+B+C+D) – 1] (e – d) '(7)LetusnowintroduceacontinuousfunctionF: (0,0)4→ (0,0) which is defined by6.bu(du1 – euz)(8)F(uo, U1, U2, U3) = Auo + Buj +Cu2 +Du3 +provided du # eu2. Consequently, we getƏF(x,x,X,X)= A= Po,On eе (А+В+С+D) -1](е — d)ƏF(x,X,X,X)= B-= P1,|Ine(9)ƏF(x,x,x,X)e[(A+B+C+D) –1](e – d)= C+2neƏF(x,x,X,X) – D=P3,En ewhere e+ d. Thus, the linearized equation of Eq.(1) aboutx takes the formZn+1- Pozn- P1Zn–k- P2Zn–1- P3Zn-o = 0,(10)where po, P1, P2 and P3 are given by (9). The objective of this article is to investigate somequalitative behavior of the solutions of the nonlineardifference equationbxn–k[dxp-k– exp-1]Xn+1 =Axn+ Bxn–k+ Cxn-1+Dxn-o +n= 0,1,2, ...where the coefficients A, B, C, D, b, d, e E (0,0), whilek, 1 and o are positive integers. The initial conditionsX-6.……., X_1,..., X_k ….., X_1, Xo are arbitrary positive realnumbers such that k < 1< o. Note that the special casesof Eq.(1) have been studied in [1] when B=C= D=0,and k=0,1=1, b is replaced by – b and in [27] whenB= C= D=0, and k= 0, b is replaced by – b and in[33] when B = C = D = 0, 1 = 0 and in [32] whenA=C=D=0, 1=0, b is replaced by – b.(1)%3D%3D

Given: xn+1=Axn+Bxn-k+Cxn-l+Dxn-σ+bxn-kdxn-k-exn-l, n=0,1,2,⋯ ...... (1) x~ is…

The equilibrium point of Eq.(1) is the positive solution of the equation X= (A+B+C+D)x+ (6) (d– e)x where d + e. If [(A+B+C+D) – 1] (e – d) > 0, then the only positive equilibrium point ž of Eq. (1) is given by b (7) [(A+B+C+D) – 1] (e – d)

The equilibrium point of Eq.(1) is the positive solution of the equation X= (A+B+C+D)x+ (6) (d– e)x where d + e. If [(A+B+C+D) – 1] (e – d) > 0, then the only positive equilibrium point ž of Eq. (1) is given by b (7) [(A+B+C+D) – 1] (e – d)

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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