n-1 ( fei+29 + fei+1k Sei-1p+ foi-ah) foi+19 + foik fesg + fei-1k ) Tau-19+ fei-2k ) hII fep+ fei-1h %3D n-1 Joi+ap + foi+2h ( fei+2P + fei+zħ\ ( feis49+ fei+ak) fei+1P+ foih ) feie+39 + fes+2k ) ( foi+6p + foi+sh\ ( foi+29 + foi+ik + foirsh, n-1 Iên-2 = r n-1 PII fei+19 + foik n-1 (foi+4P+ foi+sh) ( Soreg + foirsk Soi+3p+ fei+2h) fei+59 + fei+ak) n-1 fei-sP+ feiezh) ( fei+s9 + foi+sk Jei+7P+ feisch) (Ffei+39 + fei+2k, 2p +h II Tên+1 = r P+h where z-4 = h, z-3 = k, r-2 = r, 1-1 = p, ro = q, {fm}- = {1,0,1,1,2,3, 5,8, ..}. Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n – 2. That is; kT foisap + fai-ah( Jaug + fei-ik Soi+3p+ fei+zh, n-2 Sei-19+ foi-ak) n-2 fei+2p + fei+zh) (feir49 + fesuak foi+1p + foih ) foi+39 + fei+zk ( fei+6P + fei+sh\ Sei+sp+ fei+ch i-0 n-2 fei+29+ fei+1k) fai+19 + Sosk n-2 ( fei+aP+ fei+sh ( feso9 + foi+sk = 9T Soi+ap+ fei+2h) Iên-6 Seins9 + fes+ek ) fei+sp+ fei+zh ( fei+sg + foi+sk fei+7p + Sos+sh n-2 2p+. Tên-5 = P+h Soi+39 + fes+2k,
n-1 ( fei+29 + fei+1k Sei-1p+ foi-ah) foi+19 + foik fesg + fei-1k ) Tau-19+ fei-2k ) hII fep+ fei-1h %3D n-1 Joi+ap + foi+2h ( fei+2P + fei+zħ\ ( feis49+ fei+ak) fei+1P+ foih ) feie+39 + fes+2k ) ( foi+6p + foi+sh\ ( foi+29 + foi+ik + foirsh, n-1 Iên-2 = r n-1 PII fei+19 + foik n-1 (foi+4P+ foi+sh) ( Soreg + foirsk Soi+3p+ fei+2h) fei+59 + fei+ak) n-1 fei-sP+ feiezh) ( fei+s9 + foi+sk Jei+7P+ feisch) (Ffei+39 + fei+2k, 2p +h II Tên+1 = r P+h where z-4 = h, z-3 = k, r-2 = r, 1-1 = p, ro = q, {fm}- = {1,0,1,1,2,3, 5,8, ..}. Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n – 2. That is; kT foisap + fai-ah( Jaug + fei-ik Soi+3p+ fei+zh, n-2 Sei-19+ foi-ak) n-2 fei+2p + fei+zh) (feir49 + fesuak foi+1p + foih ) foi+39 + fei+zk ( fei+6P + fei+sh\ Sei+sp+ fei+ch i-0 n-2 fei+29+ fei+1k) fai+19 + Sosk n-2 ( fei+aP+ fei+sh ( feso9 + foi+sk = 9T Soi+ap+ fei+2h) Iên-6 Seins9 + fes+ek ) fei+sp+ fei+zh ( fei+sg + foi+sk fei+7p + Sos+sh n-2 2p+. Tên-5 = P+h Soi+39 + fes+2k,
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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Show me the steps of determine blue and information is here step by step
Transcribed Image Text:Br,-10n-2
YIn-1 + 8xn-4'
In+1 = QI,n-2 +
n = 0, 1,...,
(1)
1
The following special case of Eq.(1) has been studied
In-1Tn-2
In+1 = In-2 +
(8)
In-1 + In-4
where the initial conditions r-4, x-3, T-2, r-1,and ro are arbitrary non zero real
numbers.
Theorem 4. Let {rn}-4 be a solution of Eq.(8). Then for n = 0, 1,2, ...
Ton-4 = hT foip + foi-1h
foi-1p + foi-ah)
( fei+29 + Soi+2k
foi+19 + ferk
n-1
n-1
( foisap + foi+ah
foi+3P+ fei+2h) ( fei-19 + fei-2k)
fe:9 + fei-1k
I'ên-3 = kI
i=0
T( foi-2p + foi+h) ( feir49 + fei+zk
fei+39 + foi+2k)
fei+29 + fei+1k).
fei+19 + fesk
Ten-2 =
foi+1P + foih
foi+ep + fei+sh
Soi+sp + fei+sh ,
( foi+ap + foi+ah ( foi+69 + foi+sk).
Sei+ap + fei+2h) fei+59 + fei+ak )
(2p + h (fei+sP+ foi+zh (feit49 + fei+3k
n-1
p+h)
th II(a-p+farch) (Fars9 + Sa+zk )
where r-4 = h, x-3 = k, x-2 = r, -1 = P, xo = q, {fm}- = {1,0,1, 1,2, 3, 5, 8, ...}.
Proof: For n = 0 the result holds. Now suppose that n >0 and that our assumption
holds for n – 2. That is;
kT (foisap + fei-3ah ( feg + fei-ik )
Soi+3p + foi+2h) (Toi-19 + foi-2k )
fei+2P + fei+1h
( fei+1p + feih ) \Foi+39 + fois+2k)
fei+ep+ fei+sh ( fei+29 + foi+ik
Sei+sp + fei+gh
Ten-9 =
i-0
Sei+49 + fei+3k
n-2
foi+19 + Sauk )
(foisap + fo+3gh ( fei+69 + foi-sk)
foi+ap + fei+zh ) Jei459 + foitak)
I6n-6 =
(2p +4) T(foi-ap + fes-zh) (foisaq + foi+sk
Jei+7p+ fei+ch) Joi+39 + fei+2k )
I6n-5 =
Now, it follows from Eq.(8) that
Z6n-4 = T6n-7 +
T6n-6 + 2ôn-9
11
foi+6P + fei+sh\ ( Soi+29 + fei+ik
( foi+6P + fei+sh
PIIasp+ feirah )
foi+29 + foi+ik
n-2
n-2
foi+6P + foi+sh ( foi+29 + fei+ik
PIISP+ feigh) Tei+19 + feik )
n-2
paII (st fa) (2 fuk)
n-2
Seseg+ fersk
Joi-19+fer-ak
im0
pa II ( (atl)
= PTT (feisap + feirsh) (feis29 + feisak
fei+sp + fei+sh,
im0
n-2
Josreg+Sesk) (Tei-19+Se-ak
i=0
-i )-
( foi+oP+ fsi+sh`
Soi+sp + fei+ah)
foi+29 + foi+1k
fei+19 + feik
=PII
12
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