Theorem 9.If k is even and I, ở are odd positivVe integers, then Eq.(1) has prime period two solution if the condition (1–(C+D))(3e– d) < (e+ d) (A+ B) , (20) is valid provided C+ D) and
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A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
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Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
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How to deduce this equation from Equation 1 Explain to me the method. Show me the steps of determine blue and inf is here
![Theorem 9.If k is even and 1, o are odd positive integers,
then Eq. (1) has prime period two solution if the condition
(1- (C+D)) (3e- d) < (e+ d) (A+ B),
(20)
is
valid,
provided
(C+D)
< 1
and
e (1– (C+D))– d (A+B) > 0.
Proof.If k is even and 1, o are odd positive integers, then
Xn= Xp–k and Xn+1 = Xŋn–1= Xn-o. It follows from Eq.(1)
that
bQ
P= (A+B) Q+(C+D) P –
(21)
(eР — dQ)"
and
БР
Q= (A+B) P+ (C+D) Q –
(22)
(e Q– dP)'
Consequently, we get
e P – dPQ = e (A+B) PQ– d (A+B) Q + e(C+ D)P
- (C+D) dPQ– bQ,
(23)
and
e Q – dPQ= e (A+B) PQ – d (A+B) P² + e(C+ D) Q
- (C+D) dPQ– ÞP.
||
(24)
By subtracting (24) from (23), we get
b
P+Q=
(25)
[e (1 – (C+D)) –d (A+B)]'
where e (1– (C+D)) – d (A+ B) > 0. By adding (23)
and (24), we obtain
e b² (1– (C+D))
(e+d) [K1 +(A+B) [e K1 – d (A+B)]²
PQ =
(26)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ecaae78-467a-4f8b-9627-a81f9986c070%2F8cb84b11-5897-4c66-8923-1d093a9f80e1%2Fyd58j7j_processed.jpeg&w=3840&q=75)
![The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn-k
dxn-k- exn-1]
Xn+1 =
Axn+ Bx,-k+ Cx,-1+ Dxp-o
п 3 0, 1,2,.....
(1)
where the coefficients A, B, C, D, b, d, e E (0,∞), while
k, 1 and o are positive integers. The initial conditions
X_0,..., X_1,..., X_k, ..., X_1, Xo are arbitrary positive real
numbers such that k <1< 0. Note that the special cases
of Eq. (1) have been studied in [1] when B=C=D= 0,
0,1= 1,b is replaced by – b and in [27] when
B= C= D= 0, and k= 0, b is replaced by – b and in
[33] when B = C = D = 0, 1= 0 and in [32] when
A=C=D=0, 1=0, b is replaced by – b.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ecaae78-467a-4f8b-9627-a81f9986c070%2F8cb84b11-5897-4c66-8923-1d093a9f80e1%2F160lm1e_processed.jpeg&w=3840&q=75)
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