0o 7h+4 II-o 2-(a+1)t-q+1 II-, 2-(4+1)t-q/N-(3q+2) 1 ΣΠ II-o P(a+1)k-(q+1)t-q <> +1 h=0 k=1 0o 7h+5 II-, ?-(4+1)t-q +1 II-, 2-(g+1)t-g/2-(29+1) 1 ΣΠ II-, 2g+1)k-(q+1)t-q +1' -5 h-0 kー1
0o 7h+4 II-o 2-(a+1)t-q+1 II-, 2-(4+1)t-q/N-(3q+2) 1 ΣΠ II-o P(a+1)k-(q+1)t-q <> +1 h=0 k=1 0o 7h+5 II-, ?-(4+1)t-q +1 II-, 2-(g+1)t-g/2-(29+1) 1 ΣΠ II-, 2g+1)k-(q+1)t-q +1' -5 h-0 kー1
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:From the equations (7) and (8),
0o 7h+4
II-0 2-(a+1)t-q +1
2-(a+1)t-q/S2_(3q+2)
1
ΣΠ
lt=0
>
TT-o 2(g+1)k-(q+1)t-q + 1
t=0
h=0 k=1
o 7h+5
2-(q+1)t-q
II-, 2-(a+1)t-q/2-(2q+1)
+1
1
ΣΠ
IT-0 P(a+1)k-(q+1)t-q +1
t=0
h=0 k=1
t=0
thus N-(3q+2)
> N-(2q+1)•
Transcribed Image Text:In this work, we deal with the following nonlinear difference equation
Sm-(79+6)
1+II-o Pm-(a+1)t-q
Im+1
m = 0, 1, ...,
(1)
where 2-(79+6), 2-(79+5), .., 2-1, lo E (0, 00) is investigated.
(d) We can generate the following formulas:
L(79+7)n+r(q+1)+s+1
= r-7)(4+1)+s+1 (1–
(T-
(IIm=1 2-(mq+m-1
m-1)+s)/(r-7)(q+1)+s+1
1+ (IIm=1 N-(mq+tm-1)+s)
n 7h+r
1
ΣΠ
Im=1 P(a+1)t-(mq+m-1)+s +1
h=0 k=1
(e) If 2(7q+7)n+(t–1)q+t → a(t-1)q+t +0 then N(7+7)n+6q+7
+0 as n –→∞,
If 2(7g+7)n+tg+t → arg+t +0 then N(79+7)n+7q+7 → 0 as n → o. t = 1,6.
(e) Suppose that a1 = aq+2 = a2q+3 = a3q+4
= a4q+5 = a5q+6 = a6q+7 = 0. By (d), we
produce the following formulas below:
(1-Tg
II-o 2-(9+1)t-q +1
I-, 2-(9+1)t-g
lim N(7q+7)n+1
= lim 2-(7g+6)
t=0
7h
1
ΣΠ
II-o 2-(9+1)k-(q+1)t-q +1
h=0 k=1
7h
1
2-(79+6) |1–
aj =
II-o 2-(9+1)t-g
+1
-(q+1)k-(q+1)t-q
+1
h=0 k=1
t=0
7h
II-o 2-(4+1)t-q +1
II-, 2-(9+1)t-g
1
ΣΠ.
IT-0 Pa+1)k-(q+1)t-q +1
a1 = 0 =
(3)
h=0 k=1
II,N-(9+1)e-q/N-(34+2) x
1-
lim 17q+7)n+4q+5 = lim 2-(34+2)
IIL,2-(q+1)t-q +1
n00
n 7h+4
ΣΠ
1
h-o II-o S2-(4+1)k–(q+1)t-q + 1)"
II 2-(9+1)t-q/S2-(34+2) x
II-, 2-c+1)e-q + 1
asq+5 = N-(39+2)
-
n 7h+4
1
ΣΠ
II-, P-(9+1)k-(9+1)t-q + 1
h=0 k=1
0o 7h+4
I-, 2-(q+1)t-q + 1
II, 2-(9+1)e-q/S2-(34+2)
1
ΣΠ
a4g+5 = 0 =
(7)
II-, P(a+1)k-(q+1)t-q + 1'
h=0 k=1
Similarly
II,2-(+1)e-q/S2-(24+1).
II,2-(a+1)t-q +1
lim l(7q+7)n+5q+6 = lim 2-(2q+1)
n 7h+5
1
ΣΠ
IT-, 2-(9+1)k–(q+1)t-q + 1,
h=0 k-1
II,2(9+1)--q/2-(2p+1) x
IIL,2-(a+1)t-q +1
a5q+6 = 2-(29+1)
n 7h+5
ΣΠ
TI-,N-(a+1)k-(q+1)t-q +1
h=0 k=1
8
0o 7h+5
1
(8)
2II T e+1)k-(4+1)t-q +1
a5q+6 = 0 =
II,2-(9+1)t-q/SN-(24+1)
h=0 k=1
Similarly
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