Proof.If 1 is even and k, o are odd positive integers, then Xn = Xp-1 and xn+1 = Xn–k = Xn-o• that It follows from Eq.(1) bP P=(A+C) Q+ (B+D) P – (30) - (eQ– dP)' and bQ Q= (A+C) P+(B+D) Q – (31) (е Р- d@)* Consequently, we get b P+Q= (32) [d (1– (B+D)) – e (A+C)]' where d (1– (B+ D)) – e (A+C) > 0, е eb (A+C) PQ= (e+d) [K2+(A+ C)] [d K2 – e (A+C)]²" (33)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Show me the steps of determine yellow and inf is here

Theorem 10.If 1 is even and k, o are odd positive integers,
then Eq. (1) has prime period two solution if the condition
(A+C) (3e– d) < (e+d) (1– (B+ D)),
(29)
is
valid,
provided
(B+D)
1
and
d (1– (B+ D)) – e (A+ C) > 0.
-
Proof.If 1 is even and k, o are odd positive integers, then
Xn = Xp-1 and xn+1
that
Xn-k= Xn-o.
follows from Eq.(1)
bP
P= (A+C) Q+(B+ D) P –
(30
(e Q- dP)'
and
bQ
Q= (A+ C) P+(B+D) Q –
(31)
(е Р- d)*
Consequently, we get
b
P+Q=
(32)
[d (1– (B+ D)) – e (A+C)]'
where d (1– (B+ D)) – e (A+C) > 0,
e b (A+C)
(e+d) [K2+(A+C)][d K2 – e (A+C)]²
е
PQ =
(33)
where K2
Substituting (32) and (33) into (28), we get the condition
(29). Thus, the proof is now completed.O
(1 – (B+ D)), provided (B+ D) < 1.
Transcribed Image Text:Theorem 10.If 1 is even and k, o are odd positive integers, then Eq. (1) has prime period two solution if the condition (A+C) (3e– d) < (e+d) (1– (B+ D)), (29) is valid, provided (B+D) 1 and d (1– (B+ D)) – e (A+ C) > 0. - Proof.If 1 is even and k, o are odd positive integers, then Xn = Xp-1 and xn+1 that Xn-k= Xn-o. follows from Eq.(1) bP P= (A+C) Q+(B+ D) P – (30 (e Q- dP)' and bQ Q= (A+ C) P+(B+D) Q – (31) (е Р- d)* Consequently, we get b P+Q= (32) [d (1– (B+ D)) – e (A+C)]' where d (1– (B+ D)) – e (A+C) > 0, e b (A+C) (e+d) [K2+(A+C)][d K2 – e (A+C)]² е PQ = (33) where K2 Substituting (32) and (33) into (28), we get the condition (29). Thus, the proof is now completed.O (1 – (B+ D)), provided (B+ D) < 1.
Thus, we deduce that
(P+ Q)² > 4PQ.
(28)
The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn-k
[dxn-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 (0,00), while
k, 1 and o are positive integers. The initial conditions
X-6…, X_1,..., X_k, ..., X_1, Xo are arbitrary positive real
numbers such that k <1< o. Note that the special cases
of 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] when
B= C= D=0, and k= 0, b is replaced by
[33] when B = C = D = 0, 1 = 0 and in [32] when
A = C= D=0, 1= 0, b is replaced by – b.
(1)
b and in
%3D
Transcribed Image Text:Thus, we deduce that (P+ Q)² > 4PQ. (28) The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn-k [dxn-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 (0,00), while k, 1 and o are positive integers. The initial conditions X-6…, X_1,..., X_k, ..., X_1, Xo are arbitrary positive real numbers such that k <1< o. Note that the special cases of 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] when B= C= D=0, and k= 0, b is replaced by [33] when B = C = D = 0, 1 = 0 and in [32] when A = C= D=0, 1= 0, b is replaced by – b. (1) b and in %3D
Expert Solution
Step 1

Given:

                  xn+1=Axn+Bxn-k+Cxn-l+Dxn-σ+bxn-kdxn-k-exn-l,n=0,1,2, .......... (1)

                                        P=A+CQ+B+DP-bPeQ-dPQ=A+CP+B+DQ-bQeP-dQP+Q=bd1-B+D-eA+C

To find:

                                         PQ=eb2A+Ce+dK2+A+CdK2-eA+C2

steps

Step by step

Solved in 3 steps

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,