Theorem 1. The following system of equations Un-1Vn-3 Un-30n-1 Un+1 = and vn+1 = (35) Vn-1(A+ Bun-1Vn-3) ил-1(С + Dun-з0-1)" (1-А) A and A = has a 2-periodic solution if u-3 = u-1,v-3 = v-1, u-3V-3 = u-2V-2 = C,B = D + 0. 41 42 (1-A), where A = Proof. Let u_3 = u_1,v_3 = v-1 and u-3V-3 = u_2V-2 = D in the exact equation (34a). Then С, В %3 2m-1 C2m + (1-C) Σ n-1 1=0 U4n-2 =u-2 T 2m m=0 A2m+1 + (1- A) E A 1=0 =u-2. (36)
Theorem 1. The following system of equations Un-1Vn-3 Un-30n-1 Un+1 = and vn+1 = (35) Vn-1(A+ Bun-1Vn-3) ил-1(С + Dun-з0-1)" (1-А) A and A = has a 2-periodic solution if u-3 = u-1,v-3 = v-1, u-3V-3 = u-2V-2 = C,B = D + 0. 41 42 (1-A), where A = Proof. Let u_3 = u_1,v_3 = v-1 and u-3V-3 = u_2V-2 = D in the exact equation (34a). Then С, В %3 2m-1 C2m + (1-C) Σ n-1 1=0 U4n-2 =u-2 T 2m m=0 A2m+1 + (1- A) E A 1=0 =u-2. (36)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Theorem 1. The following system of equations
Un-1Vn-3
Un-3Vn-1
Un+1 =
and vn+1 =
(35)
Vn-1(A+ Bun-1Vn–3)
Un-1(C+ Dun-3Vn–1)'
(1-А)
has a 2-periodic solution if u_3 = u_1, v_3 = v_1, u_3v_3 = u_2v_2 = " and A =
42 C, B = D+ 0.
41
-A), where A = C,B =
Proof. Let u _3 = u_1,v_3 = v_1 and u_3v_3 = u_2v_2 =
D in the exact equation (34a). Then
2m-1
2m
+ (1-C) Σ C
n-1
l=0
U4n-2 =u-2 |T
2m
A2m+1 + (1– A) E A!
m=0
1=0
=u-2.
(36)
![In [3], the author determined and formulated the analytical solutions of the rational
recursive equations:
Xn-1Yn-3
Yn-1Xn-3
Xn+1 =
Уп+1
(2)
Yn-1(±1+ Xp-1Yn-3)'
Xn-1(+1+ yn-1Xn-3)'
here x-3, x-2, x-1, X0, y-3, y-2, y-1 and yo are the initial conditions which are arbi-
trary non-zero real numbers.
The aim of this study is to generalize the results in [2,3] by studying the system of
ordinary difference equations
10
Un-10n-k-1
Vn–k+1(An + Bnun-10n-k-1)'
Un-k-1Vn-1
Un+1
Un+1 =
(3)
Un-k+1(Cn + DnUn-k-1Vn–1)'
where An, Bn, Cn and D, are real sequences, using a symmetry method. For a similar
11
Theorem 1. The following system of equations
Un-1Vn-3
and Un+1
Un-30n-1
(35)
Un+1 =
Vn-1(A + Виm-10n-3)
Un-1(C+ Dun-3Un–1)'
(1-A) and A =
has a 2-periodic solution if u_3
С, В 3 D # 0.
= u-1, V-3 = v_1, U_3V-3 = u-2V-2 =
41
42
(1-A)
A), where A = C,B =
Proof. Let u _3 = u_1,0_3 = v_1 and u-3v-3 = u-20_2 =
D in the exact equation (34a). Then
2m-1
C2m
+ (1-C) ΣC
1=0
n-1
U4n-2 =u-2T
2m
m=0 A2m+1 + (1 – A) E A'
1=0
=u-2.
(36)
Following the same procedure as above on equations (34b) through (34h) gives u4n–1=
u-1; U4n = uo; U4n+1 = U1 and v4n-1 = v_1;V4n = Vo; V4n+1 = V1.
43
44
2т-1
C2m + Du -2vo E C
1=0
n-1
U4n-2 = U-2 ||
(34a)
u-200
2m
m=0
A2m+1 + Buov_2 E A
2m-1
C2m
+ Du-1v1 E c
410-1
u-101
I=0
U4n-1 = u1
(34b)
2m
m=0
A2m+1
+ Buη_1 Σ Α'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d6994c6-9c0b-4312-919e-29acb733a883%2Fd8b8ab10-c21a-4640-a7c3-189d62d98708%2Fymvnmi_processed.png&w=3840&q=75)
Transcribed Image Text:In [3], the author determined and formulated the analytical solutions of the rational
recursive equations:
Xn-1Yn-3
Yn-1Xn-3
Xn+1 =
Уп+1
(2)
Yn-1(±1+ Xp-1Yn-3)'
Xn-1(+1+ yn-1Xn-3)'
here x-3, x-2, x-1, X0, y-3, y-2, y-1 and yo are the initial conditions which are arbi-
trary non-zero real numbers.
The aim of this study is to generalize the results in [2,3] by studying the system of
ordinary difference equations
10
Un-10n-k-1
Vn–k+1(An + Bnun-10n-k-1)'
Un-k-1Vn-1
Un+1
Un+1 =
(3)
Un-k+1(Cn + DnUn-k-1Vn–1)'
where An, Bn, Cn and D, are real sequences, using a symmetry method. For a similar
11
Theorem 1. The following system of equations
Un-1Vn-3
and Un+1
Un-30n-1
(35)
Un+1 =
Vn-1(A + Виm-10n-3)
Un-1(C+ Dun-3Un–1)'
(1-A) and A =
has a 2-periodic solution if u_3
С, В 3 D # 0.
= u-1, V-3 = v_1, U_3V-3 = u-2V-2 =
41
42
(1-A)
A), where A = C,B =
Proof. Let u _3 = u_1,0_3 = v_1 and u-3v-3 = u-20_2 =
D in the exact equation (34a). Then
2m-1
C2m
+ (1-C) ΣC
1=0
n-1
U4n-2 =u-2T
2m
m=0 A2m+1 + (1 – A) E A'
1=0
=u-2.
(36)
Following the same procedure as above on equations (34b) through (34h) gives u4n–1=
u-1; U4n = uo; U4n+1 = U1 and v4n-1 = v_1;V4n = Vo; V4n+1 = V1.
43
44
2т-1
C2m + Du -2vo E C
1=0
n-1
U4n-2 = U-2 ||
(34a)
u-200
2m
m=0
A2m+1 + Buov_2 E A
2m-1
C2m
+ Du-1v1 E c
410-1
u-101
I=0
U4n-1 = u1
(34b)
2m
m=0
A2m+1
+ Buη_1 Σ Α'
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