In this case, system (1.8) is expressed as (af (Xn-1)+bf (*n-2)), Yn+1 = f (af (yn-1)+bf (yn-2)), (2.1) Xn+1 = for n e No. Since f is "1 – 1", from (2.1) f (Xn+1) = af (Xn-1)+bf (x,-2), f(Vn+1) = af (yn-1)+bf (yn-2), (2.2) for n E No. By using the change of variables f (xn) = un, and f (yn) = Vn, n>-2, (2.3) system (2.2) is transformed to the following one Un+1 = aun-1 + bun-2, Vn+1 = avn-1+ bvn-2, (2.4) for n e No. By taking a = 0, b= a, c= b in (1.4) and S, = Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), the solutions to equations in (2.4) are given by Un = uoJn+1+u-1Jn+2+ bu_2Jn, (2.5) Vn = VOJN+1+v-1Jn+2+bv_2Jn; (2.6) for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system (2.2) is given by Xn = f(f (x0) Jn+1+f (x-1)Jn+2+bf (x-2) Jn), n> –2, Yn = f (f (vo) Jn+1+f (y-1)Jn+2+bf (y-2) Jn), n> -2. (2.7) (2.8) 2.2. Case 2: Pn = Xn, qn = Xn, I'n = Xn, Sn = Xn In this case, system (1.8) becomes Xn+1 =f(af (Xn-1)+bf (xn-2)), Yn+1 =f(af (xn-1)+bf (xn-2)), (2.9) for n E No. It should be first note that from the equations in (2.9) it immediately follows that x, = yn, for all n EN. From (2.7), the general solution to system (2.9) is Xn = Yn = f(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n € N. (2.10)
In this case, system (1.8) is expressed as (af (Xn-1)+bf (*n-2)), Yn+1 = f (af (yn-1)+bf (yn-2)), (2.1) Xn+1 = for n e No. Since f is "1 – 1", from (2.1) f (Xn+1) = af (Xn-1)+bf (x,-2), f(Vn+1) = af (yn-1)+bf (yn-2), (2.2) for n E No. By using the change of variables f (xn) = un, and f (yn) = Vn, n>-2, (2.3) system (2.2) is transformed to the following one Un+1 = aun-1 + bun-2, Vn+1 = avn-1+ bvn-2, (2.4) for n e No. By taking a = 0, b= a, c= b in (1.4) and S, = Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), the solutions to equations in (2.4) are given by Un = uoJn+1+u-1Jn+2+ bu_2Jn, (2.5) Vn = VOJN+1+v-1Jn+2+bv_2Jn; (2.6) for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system (2.2) is given by Xn = f(f (x0) Jn+1+f (x-1)Jn+2+bf (x-2) Jn), n> –2, Yn = f (f (vo) Jn+1+f (y-1)Jn+2+bf (y-2) Jn), n> -2. (2.7) (2.8) 2.2. Case 2: Pn = Xn, qn = Xn, I'n = Xn, Sn = Xn In this case, system (1.8) becomes Xn+1 =f(af (Xn-1)+bf (xn-2)), Yn+1 =f(af (xn-1)+bf (xn-2)), (2.9) for n E No. It should be first note that from the equations in (2.9) it immediately follows that x, = yn, for all n EN. From (2.7), the general solution to system (2.9) is Xn = Yn = f(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n € N. (2.10)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In this case, system (1.8) is expressed as
Xn+1 = f (af (xn-1) +bf (xp-2)), yn+1 =ƒ'(af (ya-1) +bf (yn-2)), (2.1)
for n E No. Since f is "1 – 1", from (2.1)
f (Xn+1) = af (xn-1) +bf (xn-2), ƒ (Vn+1) = af (Yn-1)+bf (yn-2),
(2.2)
for n e No. By using the change of variables
f (xn)
= Un, and f (yn) =
= Vn, n> -2,
(2.3)
system (2.2) is transformed to the following one
Un+1 = aun-1+bun-2, Vn+1 =avn-1+bvn–2,
(2.4)
for n E No. By taking a = 0, b=a, c=b in (1.4) and S, = Jn+1, for all n > -2,
which is called generalized Padovan sequence, in (1.5), the solutions to equations in
(2.4) are given by
Un = uoJn+1+u_jJn+2+bu_2Jn,
(2.5)
Vn = VoJn+1 +v_jJn+2+bv_2Jn;
(2.6)
for n e No. From (2.3), (2.5) and (2.6), it follows that the general solution to system
(2.2) is given by
Xn = f(f (x0) Jn+1+f (x=1)Jn+2+bf (x-2)Jn), n> -2,
Yn = f(f (yo) Jn+1+f (y-1)Jn+2+bf (y-2) Jn), n2 -2.
(2.7)
(2.8)
2.2. Сase 2: pn — Хп, qn — Xп, Гn — Xn, Sn — Xn
In this case, system (1.8) becomes
Xn+l=f (af (xn-1)+bf (xp–2)), Yn+1=f"(af (xn-1)+bf (xp-2)), (2.9)
for n e No. It should be first note that from the equations in (2.9) it immediately
follows that x, = yn, for all n E N. From (2.7), the general solution to system (2.9) is
Xn = Yn = f¯' (f (xo) Jn+1+f (x_1)Jn+2+bf (x_2)Jn), nE N.
(2.10)
Transcribed Image Text:The equation
axn-Xn-k
Xn+1 =
nE No,
(1.1)
bxn-p±cXn-q
where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non-
negative integers and a, b, c are positive constants, is one of the difference equations
whose solutions are associated with number sequences. Positive solutions of concrete
Motivated by this line of investigations, here we show that the systems of differ-
ence equations
Xn+1 =f'(af (Pn-1)+bf (qn-2)), yn+1 =f'(af (rn-1)+bf (Sn-2)), (1.8)
for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences xn and
Yn, f : Df → R is a "1 – 1" continuous function on its domain Df CR, the initial
values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b
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