Question

3. Solve differential equation by undetermined coefficient methods y" + 2y' +2y = 5 3 4. Solve differential equation by undetermined coefficient methods 1 + 6y +8y = 3 - 2 + 2.

Answer 1

Ef 62 co) Given differential equation Y"+ay'tay=se duxilianty equations for homogeneous m2 +2m +2 = 0 m = -2 Iſu-8 -2 IV-Y solution ig given by -2 +22 N ante 2 2 :: yc = ( cos x + 6 sinx de Herce flan Seor, so the only Es ex so let Yp = Aeon (2) cen-determined function yo" = 366 267 ; enu) putting reget 62 7) Yp = 6ACSK 36466*+18 » 50 ACER = 506 42A0ER 5e6n 62 > A=5 11 ito 62 ,, yp 10 So The general solutions y = yptic > y = (c, cosnt Qsinxjen + 62 Ans ) 10 (1) M= 4) Gyéven differential equation is o y" +6y'+8y = 30*2*+ 2x Suxiliary equation lly_m%46+8=0 -612 -4,-2 :: Y = Cie fem2 = 30-32728 so.cindetermined functions are ė So the undermined) set fore 8-21 s 30-273 and for x & {x,13 -61V 36-32 2 " -4% tge-21 (2). Here -2x and a

-22 tchto -22 -22 -22 so Let Up = Axe22 + Be => y = Me 29-242622-23€ +C = (A-2B) e 2 ane te = yo = -261-2B)227-2462274AKE-2X 4B44) = (4B - 4A)e-2x +41xé Now putting in a cue get. (43-44 je? + 4Axe 22 -2A22% €] +8[amezit Be-227c2+p] = 3e-72+28 → [4B - 4A +6A-12B +837e-22 + [44–12A+8Abee?" +(6C +8D) +80 x = 36-22 +20 Comparing the co-efficients of ei xe 2x, x o get 10) JA = ₂ +6[A-2B)e-22 -2a ue 2A = 3 8C = 2 ic I 4. 6(+8D = 0 I SO 8D = -60 3 16 1) D 3 16 -6(6) » D = - 6 / - 3 Buel we can't calulale B. So let B remain of it is. - Up = 3 xe 22 - 4 -2x + Be 3 + u 16

Faxing B=1 ul -22 t 16 get typ - 32624 2 - 3 solution y = Jc typ. => y = ciete +(3x+1) 82% O o; General tge 22 (please like if you love this answer Aus)