Question

3) Resuelve las siguientes ecuaciones diferenciales. a) y(5) – 2y4) + 17y""' = 0

Answer 1

3 sol: - Given differentia 1 y5_244 +11y"=0 Characteristic Egn of O - 0 is 25-284 +1783 = 0 3 (72-27 +17) = 0 2-2 r +1750 which is quadratic Egh ax?+bx+c=0 where a -1, b = -2, (=17 " r = -b+ 562-490 re - (-2) + 16294-4 (1303) ca 2013 r = 2 + √ 4 - 69 8= 2 1 5-64 -IEL 64 Y 2 +81 Yux (141) = 1 +41 .. Y -0,0,0, 141 General solution y = C1 + I (2 + 22213 4 Cos 4x + (5 sin que Te Since Equation ① is homogeneous Eqn there fore y=Y.

6801:- y"+y'-64 = 4x-2 form 3 4"=0y, y'= Dy write in operator (D²+D-634 = 40-2 ③ is of the form Eqn F(D)y - Qin) where FCD] =D²+1-6. Oral = 47-2 e baracteristic Egh na 72 +1-6=0 gy2+37-276=0 (r+3) -2 (V+3) = 0 6-2) (~+3) =0 T=2, r=-3 Complementary function Me = c e2k trz e3u particular solution Up = 2 FD) y = 47-2 D?+1-6 4-7- -6(1-(9240) = 42-2 (1-(04400) = 44-2 (1+ (0030) +(232207 --> grans [ 4-2 + +0] -30 = - = [44-] D%C4*)=0 03 (4a) = 0 yp

42 Yp = - 40 + 8 yp = -21 + 2 General solution y = yctyp © © Given y"+y = cos se write in operator form (12+12y = cos x + Equis of the form F(Diya where f(1) = 77 Cosa - !+ Cosen y"Egly Qlz) = 1+ cos2a 2 Characteristic Eqw g2+1=0 = r² = -1 Yeti Complementary function Ye = ci cosa + cz sion] Particular solution Up = a F(D) yp = 1+ COS24 D2+1 yozileon & 2 02+1) Cosaa + em can t Herto fol f(a) Ip = lrt Cosan 0'+1 2+1 .. Gosbu - Cosbe f66") f(-5) if f(-5°) 70

- 4 +1 Yr = { [ { + Cosza] = { [ ! + cos2x] General solution y=9ctyp y = c, cosx + 62 Sina + - Cosza Ginen xty tny 444 =0 write in operator form (?+ OLD + 4) y co © which is cauchy Enter Equation homogeneous Equation Let a=e, 2=logn du nyo dz Xo-o a D2 - 0(0-1) e substitute in Equ © (OCO-1) +0+4) y =0 102_8+0+4 ] 4 = 0 10²+4] 4 =0 Characteristic Egn - g2+4=0 ar2=-4 9 4 = 421 General solution y=c, Cos 22 +6₂ Sin 22 - zalogn ly= e, Cos(a loga) + C₂ Sin (2 loga)