U8n-6 U8n-10U8n-4 = U8n-6 – U8n-8U8n-8-U8n–10 11. EXACT SOLUTION OF EQ. (2) WHEN a =B =y= 6 = 1This section shows the exact solutions of the following equation:Un-1Un-5Un+1 = Un-1-n = 0, 1, ..,(18)Un-3 + un-5where the initial conditions are selected to be positive real numbers.Theorem 9 Let {un}-5 be a solution to Eq. (18) and suppose that u-5 =a, u-4 = b, u_3 = c, u_2 = d, u-1 = e, uo = f. Then, for n = 0,1,2, ..., thesolutions of Eq. (18) are given by the following formulas:e2n 2n-1IT (ie + c)(ic +a)'f2n 2n-1IT (if + d)(id + b)c2n e2n2n-1U8n-5 =2n-1U8n-4 =2n-1U8n-3 = =IT (ic +a) IIT (ie + c)d2n f2nIT (id + b) IT, (if + d)'U8n-2 =e2n+1 2nU8n-1 =IT", (ie + c)(ic + a)fan+l2nIT, (if + d)(id + b)U8n =2nc2n+1e2n+1U8n+1 =2n+1IT (ic + a) I (ie + c)'d?n+1 f2n+1U8n+2 =T2n+1IIT" (id + b) IT (if + d)Proof. It can be easily seen that the solutions are true for n = 0. We supposethat n >0 and assume that our assumption holds for n – 1. That is,e2n-2,2n-3U8n-13 =2n-3IE1"(ie + c)(ic+ a)f2n-2d2n-3U8n-12 =T2n-3IT (if + d)(id + b)c2n-22n-2U8n-11 =T2n-22n-3II (ic + a) II,", (ie + c)d?n-2 f2n-2U8n-10 =12л -2,II (id + b) [I (if + d)e2n-1,2n-22n-3U8n-9 =II (ie + c)(ic +a)'f2n-12n-2T2n-2II", (if + d)(id + b)c2n-1e2n-1U8n-8 =U8n-7 =2n-IT", (ic + a) IT",(ie +c)dan-1 f2n-1IT (id + b) IIif + d)U8n-6 =i=1Moreover, Eq. (18) gives us thatU8n-6U8n-10U8n-4 = U8n-6U8n-8 + U8n–10d2n-1 f2n-1d2n-1f2n-1ךלקd2n-2 f2n-2T2n-3IT (id+b) I?",?(if+d) [T?,?(id+b) [T?"(if+d)f2n-1 d2n-2IT",?(iS+d)(id+b)d²n-2 f2n–2I (id + b) [I(if+d)dn-1 f2n-1IT (id + b) T",(if + d)-1+II?(id+b) 1?",³(if+d)d2n-1 2n-2T (id+b) I",(if+d) IT(if+d)(d2n-1 f2n-2dn-1 f2n-1II,"(id + b) II(if +d)2n-2=D12п-1li=1II (id + b) II(if +d) (en-7+a +d?n-1 f2n-1((2n - 2)f + d)IT (id + b) II" (if + d) ((2n – 1) f +d)(2n – 2)f + d`(2n – 1)f + d2n-2den-1 f2n-1an-1(id+b) [T (if+d)2n-1 f2n–12n-22n-12n-2i=1i=1i=1%D1-I (id + b) IT,if + d)d2n-1 f2nI, (id + b) II{if + d)((2n – 1)f + d)f2n ď²n-12n-'(if + d)(id +b)

We use given terms to find required term.

Answered: U8n-6 U8n-10 U8n-4 = W8n-6 U8n-8 -…

U8n-6 U8n-10 U8n-4 = W8n-6 U8n-8 - U8n-8-U8n-10

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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