Question

(27 points) Find the general solution of the associated homogeneous equation for each nonhomogeneous differential equation below. Then determine the form of a particular solution y, of the nonhomogeneous equation. Do not solve for the undetermined coefficients in yp: (a) (10 points) y" - 9y' - 22y = 5xe-2x (b) (10 points) y" – 4y' + 29 y = 8x sin 3x

Answer 1

a) y" - syl-22y = 50e2 Then auxilary equation is m2 -9M-22=0 (m-1 ) cm+2) - mall, -2 So ge=g e'r + C 24 Then Ip (suppose f(u) = oma Then uportes D2-ID-22 () where a q are constant let yo be particular sebuton. (5x02%) (suppose emmy (-2)2-9CD-2)-22 Fo+m) (x) D²-40 +4.00 - if fim) 70 & F(D+m) to D2-131 = 5ē zu MAI (W 10 5 8 20 () - ēda . T3D -20 - se 13 e2u ó [+ B] CV) 15) ( a +9) 16 ht 13 s- e2u There fane requined solution is general + q y = ye typ= q ela 92 ēzuen ēzu ( + 73.

b) = 84 sin(3x) yl-4yl tasy Then auxilang au xilang equation is is m2_4m 429 mo 4 & 116 116 2 = 47 101 2 OS & (nEhis 32551 ye = gelu cos (58) + 20 singu Now dP D?_4D+29 D2-40 +29 D2=4D+29 182.elson -izu 2 i suason 21 1 -1.4 Bu (u) tin ise (20) 132 Echie (D+3)=-4(0431) +29 [10-1372-410-3i) +28] (1) thi pia i [12+601-9 -40 - 120i + 28] D2- 6Di-9-40+121 40+121728] (29 Hi pisu 34 + 6pi-4D #20-07)/() +43 palisa risol D2-6D-4D +620412 2012:30 I + (6 Dicap (20-121 + hi e 34 20-121 120-121)

Linear algebra -4i 83x (20-121) GD1-4D+D2 (208=121) (1 +41 zisa 120+12;) D2 – G DI-4D (0) pisany 20+121 -hi ex lio (6 Di-4D + D2) 27 (20-121) (20-121) 4123 (919 [ to - (D260i-4p) (20+121) (20+12:34 (02 einn & 5.-30) ein ut. - (5+31 J Therefore if 543.) ) 30 + 34 -1 (5+3i) ne (5531); else ye 34 - 15:42') % (cos3x4 tisin3x) + (5-3/)} .mn (@ossmcisin3%) 2 34 34 o imaginary part of this Sucosa 34 + ** 3 ustang a) su cossu 3rsinsx) fina)] 34 [ 3 usingan pertany 20.30 tot in sin 3 e a mente COS f 14 da sine a) 39 Cos(3x) Therefore the nequined solution is y2= GM co s(5x) + ez zu sinGuy + un sin(3x) 17 + 2 cos(3x)