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Yn+1 = ayn-p+YYn-h (yn-h) 9-¹+8

Yn+1 = ayn-p+YYn-h (yn-h) 9-¹+8

Advanced Engineering Mathematics
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 red In the same way as proof of Theorem 2 in the second picture and inf is here

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