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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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Show me the steps of determine yellow and all information is here
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
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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