16n-4 = X6n-3 = n-1 hII i=0 n-1 k II i=0 I6n = X6n-2 = Т II (Jei+2P + fei+1h + foih i=0 n-1 X6n-1 = P PII i=0 n-1 ¶ II i=0 X6n+1 = T f6ip+f6i-1h f6i-1p+ foi-2h 5) (Sol+ (f6i+4P+f6i+3h) f6iq+f6i-1k h) (fe f6i+3P+f6i+2h, f6i-19+ foi-2k, fai+2p+f6i+1h (f6i+49 +f6i+3k\ f6i+39+f6i+2k foi+29+f6i+1k` f6i+19+ foik f6i+6p+f6i+5h f6i+5p+f6i+4h, f6i+4p+f6i+3h) foi+3p+f6i+2h, 2p+h p+h f6i+29+f6i+1k foi+19+ feik II i=0 (f6i+69+f6i+5k` f6i+59 + f6i+4k (fot f6i+8p+f6i+7h foi+7P + foi+sh f6i+49 + foi+3k f6i+39 + foi+2k
16n-4 = X6n-3 = n-1 hII i=0 n-1 k II i=0 I6n = X6n-2 = Т II (Jei+2P + fei+1h + foih i=0 n-1 X6n-1 = P PII i=0 n-1 ¶ II i=0 X6n+1 = T f6ip+f6i-1h f6i-1p+ foi-2h 5) (Sol+ (f6i+4P+f6i+3h) f6iq+f6i-1k h) (fe f6i+3P+f6i+2h, f6i-19+ foi-2k, fai+2p+f6i+1h (f6i+49 +f6i+3k\ f6i+39+f6i+2k foi+29+f6i+1k` f6i+19+ foik f6i+6p+f6i+5h f6i+5p+f6i+4h, f6i+4p+f6i+3h) foi+3p+f6i+2h, 2p+h p+h f6i+29+f6i+1k foi+19+ feik II i=0 (f6i+69+f6i+5k` f6i+59 + f6i+4k (fot f6i+8p+f6i+7h foi+7P + foi+sh f6i+49 + foi+3k f6i+39 + foi+2k
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
How was the determine blue equation deduced from the (8) equation?
Transcribed Image Text:Brn-1Tn-2
YIn-1 + 8xn-4
In+1 = aI,n-2 +
n = 0,1, ...,
(1)
The following special case of Eq.(1) has been studied
Tn-1Tn-2
In+1 = In-2+
(8)
In-1+ In-4
where the initial conditions I-4, x-3, -2, -1,and ro are arbitrary non zero real
numbers.
Theorem 4. Let {In}-4 be a solution of Eq.(8). Then forn=0,1,2, ...
п-1
feip + fei-ih
fei+29 + fei+1k
X6n-4
fei-1p+ fei-2h) foi+19 + feik
i=0
п-1
fei+4P+ fei+3h
k1I(Fei43p + fei+2h
feig + fei-1k
(fei-19 + fei-2k)
X6n-3
i=0
n-1
П
fei+2P + fei+1h ( fei+49 + fei+3k
föi+1p + feih
X6n-2
foi+39 + fei+2k,
i=0
п-1
( foi+6P+ föi+5h\ ( fei+29 + foi+ik
PII
foi+5P + fei+ah
X6n-1
%3D
fei+14 + foik
i=0
n-1
(fei+4p+ foi+3h
fei+3P + fei+2h) \ foi+59 + foi+ak,
foi+69+ fei+sk
X6n
i=0
n-1
(fei+8P+ fei+7h
Jei+7P+ foi+sh) foi+39 + fei+2k)*
2p + h'
fei+49 + foi+3k
X6n+1
p+h
i=0
where r-4 = h, x-3 = k, x-2 = r, x-1 p, xo = q, {fm}m=-1
Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption
holds for n - 2. That is;
{1,0, 1, 1, 2, 3, 5, 8, ...}.
п-2
foi+4p + foi+3h
kII
foi+3P + foi+2h,
feig + foi-1k
foi-19+ foi-2k)
X6n-9
%3D
i=0
п-2
I.
(fei+2P+ fei+1h
foi+1p + foih )
foi+49+ fei+3k
\Soi+39+ foi+2k)
X6n-8
i=0
п-2
(fei+6P+ fei+sh\ ( fei+29 + fei+1k
foi+5P + foi+ah)
X6n-7
foi+19 + foik
i=0
n-2
fei+4P+ foi+3h
foi+3P + foi+2h) (fei+59+ foi+ak,
foi+69 + fei+sk
X6n-6
i=0
n-2
(2р +h
fei+49 + fei+3k
()IIap + feneh ) (Feir3q + fai+ak)
T6n-5
%3D
p+h
foi+7p+ foi+6h
foi+39 + fei+2k )
i=0
Now, it follows from Eq.(8) that
X6n-626n-7
X6n-4 = X6n-7+
X6n-6 + X6n-9
11
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
