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Xn+1 axn_b+Yxn-k -1 (xn−b)ª−¹+y

Xn+1 axn_b+Yxn-k -1 (xn−b)ª−¹+y

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 pink In the same way as Theorem 7 in the second picture and inf is here

Xn-k In+1 = (1) (xn-k)"- +a where the initial values a-k+! = a-k+! ,l = 0,1,2, ..k. are nonzero real numbers and with (x-k+1)4-1 -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 . In the begining , we try to deduce the general solution for the following nonlinear rational generalized quadratic difference equations of order one in the form : Xn+1 = (2) (xn)? +a where (ro)? + -a Theorem 7. 1) The equilibrium point T = 0 is locally asymptotically stable if Ja| > 1. 2) The equilibrium points T =±VT– aare locally asymptotically stable if 0 < a <1. 3) All solutions of equation (2) at the equilibrium points 0 and ±V1 - aure unstable if -1< a <0 Proof. let f : (0, 0) → (0, 0) be a continuous function defined by f(u) u? +a It is easy see that df (u) -u? + a du (2² + a)? At the equilibrium point = 0, we have df (u), 1 =0 = = p du The corresponding linearized equation about 7 = 0 is given by Yn+1 - Pyn = 0 This implies that the characteristic equation is 1 =0 Hence the equilibrium point I = 0 is locally asymptotically stable if Ja| >1. Now we will prove the theorem at the equilibrium point a = equilibrium point = -V1- a by the same way. At the equilibrium point a = V1 - +V1- a and the proof at the a we have df (u), 2a – 1 = q du T=V1-a The corresponding linearized equation about = V1- a is given by Yn+1 TYn = 0 This implies that the characteristic equation is - (2a – 1) = Hence the equilibrium point 7 = V1- a is locally asymptotically stable if |2a – 1| <1. This means that 0 < a < 1.
Transcribed Image Text:Xn-k In+1 = (1) (xn-k)"- +a where the initial values a-k+! = a-k+! ,l = 0,1,2, ..k. are nonzero real numbers and with (x-k+1)4-1 -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 . In the begining , we try to deduce the general solution for the following nonlinear rational generalized quadratic difference equations of order one in the form : Xn+1 = (2) (xn)? +a where (ro)? + -a Theorem 7. 1) The equilibrium point T = 0 is locally asymptotically stable if Ja| > 1. 2) The equilibrium points T =±VT– aare locally asymptotically stable if 0 < a <1. 3) All solutions of equation (2) at the equilibrium points 0 and ±V1 - aure unstable if -1< a <0 Proof. let f : (0, 0) → (0, 0) be a continuous function defined by f(u) u? +a It is easy see that df (u) -u? + a du (2² + a)? At the equilibrium point = 0, we have df (u), 1 =0 = = p du The corresponding linearized equation about 7 = 0 is given by Yn+1 - Pyn = 0 This implies that the characteristic equation is 1 =0 Hence the equilibrium point I = 0 is locally asymptotically stable if Ja| >1. Now we will prove the theorem at the equilibrium point a = equilibrium point = -V1- a by the same way. At the equilibrium point a = V1 - +V1- a and the proof at the a we have df (u), 2a – 1 = q du T=V1-a The corresponding linearized equation about = V1- a is given by Yn+1 TYn = 0 This implies that the characteristic equation is - (2a – 1) = Hence the equilibrium point 7 = V1- a is locally asymptotically stable if |2a – 1| <1. This means that 0 < a < 1.
axn-b+yxn-k Xn+1 (an-6)9–1 Solve for xerror
Transcribed Image Text:axn-b+yxn-k Xn+1 (an-6)9–1 Solve for xerror
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