Yn+1 = ayn-p+YYn-h (yn-h) 9-¹+8 +6

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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Find the determine yellow In the same way as Theorem 2 in the second picture and inf is here i need th ans too much

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 .
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 .
aYn-p+YYn-h
Yn+1
(Yn-h)ª+8
Transcribed Image Text:aYn-p+YYn-h Yn+1 (Yn-h)ª+8
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