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Theorem 5. Suppose that {xn} is the solution of the difference equation (1) and the initial values ,l = 0,1, 2, ....k. are arbitrary nonzero real numbers .Let x k+l = a_k+l where X-k+l = a-k+l 1 = 0,1, 2, ..k. Then ,by using the notations(IX), the solutions of the difference equation (1) are given by: (7) a-k+l A ) – n-1 a-k+l 9-2 1(7) An' i =1 II (4ª")"* where T 1,2, ...k +1 ,1= 0,1, 2, ..k and n > 2. Proof. We can the mathematical induction as in theorems(2.1) and (2.2) , to prove this theorem .

Theorem 5. Suppose that {xn} is the solution of the difference equation (1) and the initial values ,l = 0,1, 2, ....k. are arbitrary nonzero real numbers .Let x k+l = a_k+l where X-k+l = a-k+l 1 = 0,1, 2, ..k. Then ,by using the notations(IX), the solutions of the difference equation (1) are given by: (7) a-k+l A ) – n-1 a-k+l 9-2 1(7) An' i =1 II (4ª")"* where T 1,2, ...k +1 ,1= 0,1, 2, ..k and n > 2. Proof. We can the mathematical induction as in theorems(2.1) and (2.2) , to prove this theorem .

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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As in the above manner we can generalize the notations(VII) by the following: (a-k+1)-1+ a (a-k+1) +a Consider AT tagIX (-11) q-1 9-1 (p-2 (q-1)(q-2) (a-k+1)9-1 (42)** + a %3D i=1 1,2, ..k + 1 1 = 0, 1,2, ..k and p2 3 %3D Xn-k In+1 = (1) 19-1 (xn-k) + a where the initial values r-k+! = a-k+! ,l = 0,1, 2, ..k. are nonzero real numbers and with (x-k+1)9-1 + -a for l = 0, 1, 2, ..k. . Moreover , we have studied the stability and periodicity of solutions for the generalized nonlinear rational difference equations in (1) and for some special cases . Now consider the following notations A1 = a4-1 + a A2 = a"-1 +a A- (p-2 I[49–1)(q-2) (III) Ap = = a9-1 + a A9-! p-1 i=1 where p> 3. р-2 Ap = a'II4? + aA-1 where p > 3. ((I)) i=1 In Xn+1 (3) q-1 (xn)"- + a where (ro)- # -a. Now consider the following notations Xn+1 (xn) + a where (xo) + -a In order to do this we introduce the following notations: Let xo = a Consider the following otations: Theorem 2.1 Suppose that {xn} is the solution of the difference equation (2) notations(I), the solutions of the difference equation (2) is given by: With the above a/A1 n-1 a Xn = An (II) i=1 where a and a are non-zero real numbers and n > 2. Theorem 2.2 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 In = An i =1 II 49-3 (IV) where xo = a and n > 2
Transcribed Image Text:As in the above manner we can generalize the notations(VII) by the following: (a-k+1)-1+ a (a-k+1) +a Consider AT tagIX (-11) q-1 9-1 (p-2 (q-1)(q-2) (a-k+1)9-1 (42)** + a %3D i=1 1,2, ..k + 1 1 = 0, 1,2, ..k and p2 3 %3D Xn-k In+1 = (1) 19-1 (xn-k) + a where the initial values r-k+! = a-k+! ,l = 0,1, 2, ..k. are nonzero real numbers and with (x-k+1)9-1 + -a for l = 0, 1, 2, ..k. . Moreover , we have studied the stability and periodicity of solutions for the generalized nonlinear rational difference equations in (1) and for some special cases . Now consider the following notations A1 = a4-1 + a A2 = a"-1 +a A- (p-2 I[49–1)(q-2) (III) Ap = = a9-1 + a A9-! p-1 i=1 where p> 3. р-2 Ap = a'II4? + aA-1 where p > 3. ((I)) i=1 In Xn+1 (3) q-1 (xn)"- + a where (ro)- # -a. Now consider the following notations Xn+1 (xn) + a where (xo) + -a In order to do this we introduce the following notations: Let xo = a Consider the following otations: Theorem 2.1 Suppose that {xn} is the solution of the difference equation (2) notations(I), the solutions of the difference equation (2) is given by: With the above a/A1 n-1 a Xn = An (II) i=1 where a and a are non-zero real numbers and n > 2. Theorem 2.2 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 In = An i =1 II 49-3 (IV) where xo = a and n > 2
Theorem 5. Suppose that {xn} is the solution of the difference equation (1) and the initial values x-k+l = a_k+l l = 0,1, 2, ....k. are arbitrary nonzero real numbers .Let x_k+l = a_ktI where l = 0,1, 2, ..k.Then ,by using the notations(IX), the solutions of the difference equation (1) are given by: (7) a_k+l (7) n-1 a_k+l q-2 II (4")** (T) An i =1 where T = 1, 2, ...k +1 l = 0, 1, 2, ....k and n > 2. Proof. We can the mathematical induction as in theorems(2.1) and (2.2) , to prove this theorem.
Transcribed Image Text:Theorem 5. Suppose that {xn} is the solution of the difference equation (1) and the initial values x-k+l = a_k+l l = 0,1, 2, ....k. are arbitrary nonzero real numbers .Let x_k+l = a_ktI where l = 0,1, 2, ..k.Then ,by using the notations(IX), the solutions of the difference equation (1) are given by: (7) a_k+l (7) n-1 a_k+l q-2 II (4")** (T) An i =1 where T = 1, 2, ...k +1 l = 0, 1, 2, ....k and n > 2. Proof. We can the mathematical induction as in theorems(2.1) and (2.2) , to prove this theorem.
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