11. EXACT SOLUTION OF EQ. (2) WHEN a = B = y = 8 = 1This section shows the exact solutions of the following equation:Un-1un-5Un+1 = Un-1n = 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 =а, и-4 3D 6, и-з 3D с, и-2%3D d, и-1 — е, ио 3D f. Then, for n 3 0, 1, 2, .., thesolutions of Eq. (18) are given by the following formulas:e2n 2n-1II (ie + c)(ic+a)*f2n d2n-1IT (if + d)(id + b)'c2n e2nU8n-5 =2n-U8n-4 =2n-1U8n-3 =2n-1IT (ic + a) IT (ie + c)d2n f2nU8n-2 =IT, (id + b) II if + d)=D1e2n+1c2nU8n-1 =IT, (ie + c)(ic+a)f2n+1d2nII (if + d)(id + b)'U8n =c2n+1e2n+1U8n+1 =2n+1ITT (ic+ a) II, (ie + c)d2n+1 f2n+1U8n+2 =2n+1IIT (id + b) [I, (if + d)'Proof.that n >0 and assume that our assumption holds for n- 1. That is,It can be easily seen that the solutions are true for n = 0. We supposee2n-2,2n-3U8n-13 =2n-3IT (ie + c)(ic + a)f2n-2?n-3IT(if + d)(id + b)2n-2 2n-2U8n-12 =2n-3U8n-11 =2n-2li=12n-3I (ic+a) I (ie + c)d2n-2 f2n-2U8n-10 =T2n-2IT(id + b) IT (if + d)li=1e2n-12n-2U8n-9 =2n-2ITE (ie + c)(ic +f2n-12n-2n-2U8n-8 =II(if + d)(id + b)c2n-1e2n-1U8n-7 =2n-2(ic+ a) II(ie +c)d2n-1 f2n-1IT (id + b) II (if + d)U8n-6 =In addition, Eq. (18) leads toU8n-5U8n-9U8n-3 = U8n-5U8n-7 + ug8n-92n 2n-12n-12n-2e2n c2n–1IT (ie+c)(ic+a) IT¿²(ie+c)(ic+a)li=1c2n-1 e2n-1IT"'(ic+a) [I?",?(ie+c)2n-1(ie+c)(ic +a)e2n –1 c2n–2T (ie+c)(ic+a)c2n-2e2n2n-2+e2n c2n-1'ie+c)(ic+a) I'(ie + c)(ic+a) I (ic+ a)(Ttieta) + eI!n--1li=1 (ic+a)e2n 2n-1c2n-2e2n%3DII"(ie + c)(ic + a) II(ie +c)(ic + a) (en-ijeta + )2n-1i3D1(2n–1)c+ac2n-1e2n ((2n – 1)c+a)I (ie + c)(ic +a) II (ie + c)(ic+ a) ((2n)c + a)(2n – 1)c+ a)(2n)c+ ae2n _2n-12n-12n-1=1i=1e2n c2n-1%3D1-2n-1II (ie + c)(ic+a)i=1e2n 2nc2n e2n%3D2п-1II"T (ie + c)(ic + a)((2n)c + a) II", (ic + a) II" (ie + c)2n-1Similarly, one can prove other solutions. U8n-5 U8n-9U8n-3U8n-5 – U8n-7 –U8n-7-U8n-9

We are asked to show the following equation, u8n-3=c2ne2n∏i=12nic+a∏i=12n-1ie+c Using the identity,…

Theorem 9 Let {um}=-5 be a solution to Eq. (18) and suppose that u-s = a, u-4 = b, u_3 = c, u_2 = d, u-1 = e, uo = f. Then, for n = solutions of Eq. (18) are given by the following formulas: 0, 1, 2, ..., the e2n 2n-1 (ie + c)(ic+a)' f2n2n-1 IT(if + d)(id + b)' U8n-5 = 2n-1 U8n-4 U8n-3 = 2n-1 IT, (ic + a) IT (ie + c)' dan f2n n-1 U8n-2 IT, (id + b) IT'(if +d)' e2n+12n U8n-1 ITE, (ie + c)(ic + a) f2n+!d2n IT, (if + d)(id + b)' U8n = c2n+1@2n+1 U8n+1 2n IT (ic+ a) (ie + c)' d?n+1 f2n+1 IT*(id + b) [T": (if + d) U8n+2 = i=1

Theorem 9 Let {um}=-5 be a solution to Eq. (18) and suppose that u-s = a, u-4 = b, u_3 = c, u_2 = d, u-1 = e, uo = f. Then, for n = solutions of Eq. (18) are given by the following formulas: 0, 1, 2, ..., the e2n 2n-1 (ie + c)(ic+a)' f2n2n-1 IT(if + d)(id + b)' U8n-5 = 2n-1 U8n-4 U8n-3 = 2n-1 IT, (ic + a) IT (ie + c)' dan f2n n-1 U8n-2 IT, (id + b) IT'(if +d)' e2n+12n U8n-1 ITE, (ie + c)(ic + a) f2n+!d2n IT, (if + d)(id + b)' U8n = c2n+1@2n+1 U8n+1 2n IT (ic+ a) (ie + c)' d?n+1 f2n+1 IT*(id + b) [T": (if + d) U8n+2 = i=1

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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