Show me the steps of determine green and inf is here Theorem 7 Let 0 < p < 1.Then every solutions of Eq.(8) is bounded andpersist such that1- pr1< yn < +p"-!Cı+p?-C1where C1 = p (v0 +).Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we havefrom Eq.(8)Yo> 1,1+PymY1Y11+P2Y1-m> 1.Y2Thus we obtain by induction yn > 1 for n > 1.Now we consider the other side. We have from Eq.(8)YnYn+1 = 1+p<1+ pyn.(12)2Yn-mAccording to Theorem 3, there exist a sequence yn < Un, n = 0, 1, .., where{un} satisfies(13)Un+1 = 1+ pun, n > 1,such that us = Ys,Us+1 = Ys+1, 8 € {-m, -m + 1, ..:},n > s. Therefore thesolution of the difference equation (13) is1- p"Un+ p?-C1(14)1- pwhere C1 = p( Yo+Moreover we obtain from (12) and (14),p-Yn+1- Un+1 s and p E (0, 1). So, we get yn < Um, n > s as desired. I Theorem 3 (See [8]) Let n e N, and g (n, u, v) be a decreasing function inu and v for any fixed n. Suppose that for n > no, the inequalitiesYn+1

Theorem 7 Let 0 < p < 1. Then every solutions of Eq. (8) is bounded and persist such that 1- p" n- 1< Yn < 1- P where C1 = p (y0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo 1+p 2 > 1, Y1 Y1 1+P2 Yi-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn+1 = 1+p2 Yn <1+ pyn. (12) Yn-m where According to Theorem 3, there exist a sequence yn S Un, n = 0, 1, · .., {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 E {-m, -m + 1, ·..},n > s. Therefore the solution of the difference equation (13) is 1- pr + p2-C1 1- p (14) Un =

Theorem 7 Let 0 < p < 1. Then every solutions of Eq. (8) is bounded and persist such that 1- p" n- 1< Yn < 1- P where C1 = p (y0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo 1+p 2 > 1, Y1 Y1 1+P2 Yi-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn+1 = 1+p2 Yn <1+ pyn. (12) Yn-m where According to Theorem 3, there exist a sequence yn S Un, n = 0, 1, · .., {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 E {-m, -m + 1, ·..},n > s. Therefore the solution of the difference equation (13) is 1- pr + p2-C1 1- p (14) Un =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Find et, where A = 5 3 -3 -1 -
Whta is 1+1=
6(2x + 5) = 10(4 + x) then x = Select one: a. 15 b. 5 c. 20 d. 10
Evaluate I = | 5+ 3r3 dr
find (AB)−1.
Make t the subject - hot 98t + 1080e 1080e-> -1080= 100
9.
where did the 26 and 10 come from?
Find A-B, if: 1 3 2 1 B = 2 3 2 A 1 1 1
Evaluate the following: 2. y(4)+8y"+16y=0
hイP 3I then 3 meN such that d- 634
SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +