and all informati
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
Show me the steps of determine red and all information is here
![X-1yn-3
Yn-1(+1+xn-1Yn-3)'
Yn-1Xn-3
Xn-1(F1±yn-1Xn-3)'
Xn+1
Yn+1
(2)
• 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
Bnun-10n-k-1)'
Un-k-10n-1
Un-k+1(Cn + Dnln-k-1ºn-1)'
Un+1 =
Un+1 =
(3)
Vn-k+1(An
where An, Bn, C, and D, are real sequences, using a symmetry method. For a similar
11
Definition 4. A symmetry generator of (4) is denoted by U and is given by
sk.
+ SQ
un+1
U =
Q:
+...+
(9)
dun+k-1
Q is the characteristic of the group of transformations.
27
In the above definition, S/Q(n, Un)
the shift operator.
Consider the system of ordinary difference equations of the form
Q(n +j,un+j). The operator S is known as
28
20
Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2(un, Un+2, Vn+k),
(10)
where the independent variable here is denoted by n and the dependent variables are
denoted by un, Un and their shifts.
Consider the group of transformations
(n, un, Vn) (n, ũn = un + EQ1(n, un)
+0(?), 0n = vn + eQ2(n, vn)+O(z²)).
(11)
In (11), the characteristic of the group of transformations is
Q = (Q1, Q2). The infinitesimal generator corresponds to
U = Q1ðun + Q2dvn,
(12)
where dx = . In this work, we will need the kth extension
uk = Qidu, + Q2don +s?Q1dun+2 + S?Q2dv+2 + s*Qidu+* + s*Q2dvn+*
(13)
%3D
of (12).
For the set of solutions of (10) to be mapped to itself, the following linearised symmetry
conditions
S(k+2)Q1 – uklwi = 0 and S(k+2)Q2 - uklw2 = 0,
(14)
whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+
2,0) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a
32 symmetry group.
30
31
3. Symmetries and Solutions of the System of Difference Equations (1)
33
Equivalently, equation (3) can be written as
Un+kVn
Un+k+2 =W1 =
Vn+2(an + bnUn+kUn)'
(15)
UnVn+k
Un+k+2 =w2 =
Un+2(Cn + dnunVn+k)'
where an and ba are real sequences. Applying (14) onto (15) yields,
-sk+2Q1 +
anun+kQ2
Un+2(an + byun+kOn)2
(16а)
Un+k®nS²Q2
v42(an + bnun+kUn)
= 0,
Un+2(an + bnUn+kUn)²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d6994c6-9c0b-4312-919e-29acb733a883%2F33b3f99e-37ab-406b-8553-fb2abddba79d%2F677v6u8_processed.png&w=3840&q=75)
Transcribed Image Text:X-1yn-3
Yn-1(+1+xn-1Yn-3)'
Yn-1Xn-3
Xn-1(F1±yn-1Xn-3)'
Xn+1
Yn+1
(2)
• 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
Bnun-10n-k-1)'
Un-k-10n-1
Un-k+1(Cn + Dnln-k-1ºn-1)'
Un+1 =
Un+1 =
(3)
Vn-k+1(An
where An, Bn, C, and D, are real sequences, using a symmetry method. For a similar
11
Definition 4. A symmetry generator of (4) is denoted by U and is given by
sk.
+ SQ
un+1
U =
Q:
+...+
(9)
dun+k-1
Q is the characteristic of the group of transformations.
27
In the above definition, S/Q(n, Un)
the shift operator.
Consider the system of ordinary difference equations of the form
Q(n +j,un+j). The operator S is known as
28
20
Un+k+2 = w1(un+k, Vn, Vn+2), Vn+k+2 = w2(un, Un+2, Vn+k),
(10)
where the independent variable here is denoted by n and the dependent variables are
denoted by un, Un and their shifts.
Consider the group of transformations
(n, un, Vn) (n, ũn = un + EQ1(n, un)
+0(?), 0n = vn + eQ2(n, vn)+O(z²)).
(11)
In (11), the characteristic of the group of transformations is
Q = (Q1, Q2). The infinitesimal generator corresponds to
U = Q1ðun + Q2dvn,
(12)
where dx = . In this work, we will need the kth extension
uk = Qidu, + Q2don +s?Q1dun+2 + S?Q2dv+2 + s*Qidu+* + s*Q2dvn+*
(13)
%3D
of (12).
For the set of solutions of (10) to be mapped to itself, the following linearised symmetry
conditions
S(k+2)Q1 – uklwi = 0 and S(k+2)Q2 - uklw2 = 0,
(14)
whenever (10) is true, must be satisfied. If the conditions given in(14), that is, Q;(n+k+
2,0) – uk (n;) = 0, j = 1,2, are satisfied, then the group of transformations (11) is a
32 symmetry group.
30
31
3. Symmetries and Solutions of the System of Difference Equations (1)
33
Equivalently, equation (3) can be written as
Un+kVn
Un+k+2 =W1 =
Vn+2(an + bnUn+kUn)'
(15)
UnVn+k
Un+k+2 =w2 =
Un+2(Cn + dnunVn+k)'
where an and ba are real sequences. Applying (14) onto (15) yields,
-sk+2Q1 +
anun+kQ2
Un+2(an + byun+kOn)2
(16а)
Un+k®nS²Q2
v42(an + bnun+kUn)
= 0,
Un+2(an + bnUn+kUn)²
Transcribed Image Text: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)
Un-1(C+ Dun-3Vn–1)'
(1-A)
B
has a 2-periodic solution if u_3 = u_1,v-3 = v_1, u_3v_3 = u_2V-2 =
42 C,B = D + 0.
and A =
41
where j = 0,1,2,3. For the sake of clarity, we can rewrite (33) in expanded forms as
follows:
2m-1
C2m + Du-2vo E C'
n-1
uov-2
1=0
U4n-2 = u_2 T
(34a)
u-200
2m
m=0
A2m+1 + Buov._2 E A'
l=0
(1-A), where A = C,B =
(1-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
2m-1
Cam+ (1- C) ΣC
n-1
l=0
U4n-2 =u-2||
2m
m=0 A2m+1 + (1 – A) E A'
l=0
=u-2.
(36)
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