Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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

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
See similar textbooks
icon
Related questions
Question

Show me the steps of determine green and all information is here

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)
In addition, Eq. (18) leads to U8n-5U8n-9 U8n-3 = U8n-5 И8п-7 + u8n-9 e2n 2n –1 e2n-1,2n-2 e2n c2n-1 (ie+c)(ic+a) [I",?(ie+c)(ic+a) e2n-1c2n-2 IT",?(ie+c)(ic+a) c2n-2e2n c2n-1e2n-1 II?",(ic+a) [I?",²(ie+c) 2n-1 П (е + c) (iс + а) i=D1 1. e2n c2n-1 II ie + e)(ic + a) II, "(ie + c)(ic +a) II (ic + a) (TT Geta) + eIlie- 2n-1 1 li=1 II (ic+a) i=1 cII"?(ic+a) e2n 2n-1 c2n-22n IT (ie + c)(ic +a) II'(ie +c)(ic + a) ( en-ie+a +) 2n-1 li=1 2n-1 c2n-1e2n ((2n – 1)c+a) ПТ (iе + с) (іс +а) ПТ (ie + с) (iс + а) (2m)с + а) (2n — 1)с + а (2n)с + а e2n c2n-1 2n-1 li=1 2п-1 li=1 e2n c2n-1 (1 II", (ie + c)(ic +a) 2n-1 li=1 e2n 2n c2n e2n 2n-1 Li=1 2n-1 -2n П (iе + с) (ic+ta)((2n)с + а) П (с + а) П-T (ie + c) i=1 i=1 Similarly, one can prove other solutions.
Transcribed Image Text:In addition, Eq. (18) leads to U8n-5U8n-9 U8n-3 = U8n-5 И8п-7 + u8n-9 e2n 2n –1 e2n-1,2n-2 e2n c2n-1 (ie+c)(ic+a) [I",?(ie+c)(ic+a) e2n-1c2n-2 IT",?(ie+c)(ic+a) c2n-2e2n c2n-1e2n-1 II?",(ic+a) [I?",²(ie+c) 2n-1 П (е + c) (iс + а) i=D1 1. e2n c2n-1 II ie + e)(ic + a) II, "(ie + c)(ic +a) II (ic + a) (TT Geta) + eIlie- 2n-1 1 li=1 II (ic+a) i=1 cII"?(ic+a) e2n 2n-1 c2n-22n IT (ie + c)(ic +a) II'(ie +c)(ic + a) ( en-ie+a +) 2n-1 li=1 2n-1 c2n-1e2n ((2n – 1)c+a) ПТ (iе + с) (іс +а) ПТ (ie + с) (iс + а) (2m)с + а) (2n — 1)с + а (2n)с + а e2n c2n-1 2n-1 li=1 2п-1 li=1 e2n c2n-1 (1 II", (ie + c)(ic +a) 2n-1 li=1 e2n 2n c2n e2n 2n-1 Li=1 2n-1 -2n П (iе + с) (ic+ta)((2n)с + а) П (с + а) П-T (ie + c) i=1 i=1 Similarly, one can prove other solutions.
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS