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: (n- a A1 n-1 a Xn = An i =1 II 4-2 (IV) where xo = a and n > 2 Proof. Firstly, a (х о)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. C- 2-1 а a q-2 2 = (x 1)4-1 + a -1 + a A1A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that (-- i r-1 a Xr = A, i =1 II 49 Finally we prove that the equations{IV} is true for n =r+ 1. Xr Xr+1 = +a
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: (n- a A1 n-1 a Xn = An i =1 II 4-2 (IV) where xo = a and n > 2 Proof. Firstly, a (х о)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. C- 2-1 а a q-2 2 = (x 1)4-1 + a -1 + a A1A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that (-- i r-1 a Xr = A, i =1 II 49 Finally we prove that the equations{IV} is true for n =r+ 1. Xr Xr+1 = +a
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:A more general form of difference equation (2) can take the following nonlinear rational
difference equations :
Xn+1
(xn)- +a
(3)
where (xo)9-1
Now consider the following notations
+ -a.
A1 = a9-1 + a
A2 = a"-1 + a Af-
q-1
(p-2
(q-1)(q-2)
Ap =
= a9-1
+ a
'p-1
(III)
i=1
where p > 3.
Transcribed Image Text: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
Xn
(IV)
An
i =1
where xo = a and n > 2
Proof. Firstly,
a
x 1 =
(x 0)ª-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 .
x 1
2-1
a
a A
II 42
x 2
(x 1)ª-1 + a
-1 + a
A1 A2
A2
i =1
Now suppose that the equations{IV} is true for n = r.This means that
r-1
a
q-2
Xr =
A,
i =1
Finally we prove that the equations{IV} is true for n = r + 1.
Xr+1 =
-1
+a
Expert Solution
Step 1
Given : nonlinear rational difference equation
(3)
where
Other notations :
Step by step
Solved in 4 steps
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