Question

depending on whether indefinite op de ol functions and an is never zero over this interval. Additional Exercises 22.1. Find the general solution to each of the following nonhomogeneous differential equations Use variation of parameters even if another method might seem easier. For your comve- nience, each equation is accompanied by a general solution to the corresponding homoge- neous equation a. ry" - 2xy' + 2y 3x, yn = cix + c2x2 b. y + y = cot(x) yn = ci cos(x) + C2 sin(x) cy" + 4y csc(2x) yn = ci cos(2x) + c2 sin(2x) O vald 101 d. y" - 7y' + 10y 6e3x, Yh C1e2x + czesto nosytool 3 And, again, 8 does not have to even be continuous. It just cannot have particularly bad discontinuities. + 1 set (yn. y2, ...yn), and, from the corresponding theory developed for these Wronskians (see.in Again, it should be noted that the Win these formulas is the Wronskian of the fundamental particular, theorem 13.6 on page 274), it follows that the above integrands will (theoretically at least) be nice integrable functions over our interval of interest as long as 8 and the ar's are continuous

22.1 A,C

Answer 1

22:1 xxy"- any' +2y = 80 Gernen Yn = antan? N 22 = 2n2_nhax? w(niny = 2n The wronhcian for nyo. n n² Lea ye ne are linearly independent + 20 (W) x2 untzen U (n) n il 7 u + 2nze ye . u'nthon²=0 and и cy satisfy Le 17 2 ze 2 zo'nt yo" ul = + zu u' + 220'n +22) lu 4+24 Le] 352 + 2 un +2 Lenz 3 zien? معطوة - ط 2 . + =) + 35u. 2 x²u 226 + zyn+zy6n2 3 20 u' + 220 220'n 3. n (1) n = ) 2 =-un from ☺ her 2 eln 3 =-u'n? WMM HOLINI NA 3 = 3 in 201 n -5 3 n 13 = 3 ina 34-3 201 = - 5/2 che = 3 du 2 -312 -572+1 -3/2 u 3 rea 2 3 - - 2 n -312 1-5/

Ano = 3 ñ -un²=3 u' =- 3 --3m-2-3 =-3 n du 1) -3 n 2 Ź dw - 3 ž el u = > - 3 - 3 LI NAT - 3 / +1 2 I = 6 6 n n 2 General solution is y = Gat G nr.+ 6. nt .+ 6.65.x tez)n can't on - 2 n NA = Garan = Gn+C n2+ 450 CSC an z Cosee 2x y" +49 = Sin an 9 Cosant ₂ so s8 ze Sihan ket u Cesant = ulasan +20 Sin 2n +62) usim 2n+22elemen of inat U ulceson U , e so chersen -2 the sin 2n=0 --99 Ye" -eu' sinan_yucesan -420 Schan+22624 -zu' sinan- 44062x-husik2x + 22'conna + 4 ulosn +40 ،2 ماکو 2n Cosee zu - 2U' Sin 2n+221 Ces 2n = Cosee 24 · 19

from (1) f (1) ul Cos ant e sinanto -220Casan + Cosce 2n=o aul Schoon ul el 11 Sinano Cosee an Cano Concern -20oshan -2 5hžn ul = : - Iz o' Cesan. Lesce aw li ㅗ 2 u = -x, 늦 Cosan dar 4 9 = 2 Sinan I en Sinan) er 2 General Solution is Os = G Cosza + C₂ Sinan - + C₂ Sinan & nasanth Sinani enfinan)