Yn+1 = ayn-p+YYn-h (yn-h) 9-¹+8
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
Find the determine red In the same way as proof of Theorem 2 in the second picture and inf is here
Transcribed Image Text:aYn-p+YYn-h
Yn+1
(Yn-h)ª+8
Transcribed Image Text:In
Xn+1 =
(3)
(xn)ª-1 + a
where (xo)ª¬ + -a.
Now consider the following notations
A1 = a9-1
+a
A2 = a?-1 +a Af-
(p-2
(q-1)(g-2)
Ap = a9-1
+a Ai
(III)
*p-
i=1
where p > 3.
Theorem 2. Suppose that {xn} is solutions of (3 ) and the initial value xo is an arbitrary nonzero
real number . Let xo = a .Then ,by using the notations(III), the solutions of (3) are given by:
a
A1
n-1
a
-2
In
(IV)
An
i =1
where xo = a and n > 2
Proof. Firstly,
a
x 1 =
(x o)9-1 + a
A1
Now ,by mathematical induction , we will prove that equations{IV} are true for n > 2.
In the begining we try to prove that equations{IV} are true for n=2.
1
2–1
a
a
II 42
I 2=
%3D
(x 1)ª–1 + a
(유)9-1 + a
A1 A2
A2
i =1
Now suppose that the equations{IV} is true for n = r.This means that
r-1
a
Xr =
A,
ПА
=1
Finally we prove that the equations{IV} is true for n =r +1.
Xr+1 =
.q-1
+ a
i =1
(* II Aa-1 + a
i =1
A-1
a
T-1
(q-2)(q-1)
A,{a9=1 II
i =1
+ aA1}
a
Ar+1
r-1
{a9-1 TI A9-2)(4–1)
+aA1}
i =1
which complete the proof .
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