Proof. First, we prove (i) by assuming that the constants A and B are nonzero numbers, $±V62-4 ko k1 $tV$2-4 ko k1 then |w2n| + 0. It is clear that if < 1, then » 0 as 2 otV62-4 ko k1 otV62-4 ko ki n → 0, while → 0 asn → 0 when > 1. Thus, 2 2 0+ Vo2 – 4 ko ki A 1- k1 + ko ko – k1, O - V02 – 4 koki lim w2n| lim + B n 00 n 00 2-4 ko k1 1- ki + ko 1- ко — k1 0 - V02 4 ko k1 A lim n 00 +B lim n 00 |() . (* $±V62-4 ko k1 -ki+ko 1-ko-ki) < 1, 0+V62-4 ko k1 > 1. Further, if wo = w-2 = 1-ki+ko 1-ko-ki u, then A and B are zero and we have 4 ko ki 1 – k1 + ko ko – k1 V 02 - 4 ko k1 Ф — V W2n A + B (1- kı + ko (1– ko – k1 ) Wo. The prove of (i) is done Similarly the other pronerties can be proved
Proof. First, we prove (i) by assuming that the constants A and B are nonzero numbers, $±V62-4 ko k1 $tV$2-4 ko k1 then |w2n| + 0. It is clear that if < 1, then » 0 as 2 otV62-4 ko k1 otV62-4 ko ki n → 0, while → 0 asn → 0 when > 1. Thus, 2 2 0+ Vo2 – 4 ko ki A 1- k1 + ko ko – k1, O - V02 – 4 koki lim w2n| lim + B n 00 n 00 2-4 ko k1 1- ki + ko 1- ко — k1 0 - V02 4 ko k1 A lim n 00 +B lim n 00 |() . (* $±V62-4 ko k1 -ki+ko 1-ko-ki) < 1, 0+V62-4 ko k1 > 1. Further, if wo = w-2 = 1-ki+ko 1-ko-ki u, then A and B are zero and we have 4 ko ki 1 – k1 + ko ko – k1 V 02 - 4 ko k1 Ф — V W2n A + B (1- kı + ko (1– ko – k1 ) Wo. The prove of (i) is done Similarly the other pronerties can be proved
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 11E
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Transcribed Image Text:Proof. First, we prove (i) by assuming that the constants A and B are nonzero numbers,
+V62-4 ko k1
0+V¢2-4 ko k1
then Jw2n| + 0. It is clear that if
< 1, then
→ 0 as
2
|(
t/$2 -4 ko k1
62–4 ko k1
n → 0, while
→ 0 as n → 0 when
> 1. Thus,
2
lim w2 n|
0 + V02 – 4 ko k1
A
62
4 ko ki
1- k1 + ko
lim
+ B
n 00
n 00
2
2
1 — Ко — k1
n
- 4 ko ki
4 ko ki
1 – k1 + ko
+
ф —
62 –
A lim
+B lim
2
- ko – k1
n 00
otV$2 -4 ko k1
1-k1+ko
1-ko-ki)
< 1,
2
0±V2-4 ko k1
> 1.
2
Further, if wo = w-2 =
1-k1+ko
1–ko-k1
u, then A and B are zero and we have
4² – 4 ko ki
Vø2 – 4 ko ki
1 – k1 + ko
+
W2n
A
+ B
2
1 – ko – k1
1 – k1 + ko
- ko – k1
wo.
The prove of (i) is done. Similarly, the other properties can be proved.
Corollary 1. Let {wn, zn}-2 be a solution of (4), thus
Znin=-2
a) If wo = w_2 =
1-ko-ki) H, then the solution of wn is periodic of period two.
b) If zo = z_2 =
1-vi+vo
1-v0-vi
€, then the solution of z, is periodic of period two.
(Boundedness). Every non-negative solution
of (4) is
unbounded if
Corollary 2
$±V¢2-4 ko k1
tV2 -4 vO v1
and
are greater than one.
2
Transcribed Image Text:Wn-p
p=0
Wn-p 2 zn-h
p=1
Σ
Zn-h
h=1
h=0
Wn+1
+µ and zn+1
+ E,
(4)
Zn - €
Wn - µ
where u and e are arbitrary positive real numbers with initial conditions w; and z; for i =
-2, –1,0.
(i) If A +0 and B + 0, then
() 4.
0+V2-4 ko kı
1-ky+ko
< 1,
lim w2n| =
n 00
¢tV02-4 ko kı
2
> 1.
0o,
() u and
0+V2-4 ko kı
On the other hand, if wo = w_2 =
1-k1+ko
1-ko-ki
+ 0, then A = B = 0
2
and w2n = wo = w_2.
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