Question

Find the general solution except when the exercise stipulates otherwise:

1. a) (D³ + 7D²+ 19D + 13) y =0; whenx=0, y=0, y'=2, y''=-12

b) (D³ + D² + 4D +4) y =0; when x=0, y=0, y'=-1, y''=5

c) d²x/dt² + 2b (dx/dt) +k²x = 0, k>b>0; when t=0, x=0, dx/dt = v₀

Answer 1

org y = 2 when on=0 2 = - ¢, 4 (263 – 3 C) G + 3 cq - 203 - 2 Again yo = e, ed + e 14% (66, +463) Sin 24 = (460+603)assa 46 cal+ 903) sinza -(63-92) Coszal pin 2x (1222 +503) - Cop21 (1293 y" - - 12 when anto. 12 ait 502 - 1263 Gen te-3ng 12 = C, 3 +562 - 12 ez = - 12 By OR® By @ f3 G-3--1 2Cq , 263 = -2 4 C2 -12 63 = -12 C2 - 3lz=-3. 203-21 = 1) C3 ¿, ==C2 = 0 C2 -1 2 11 41216 general solo be y = -34 Sinan

G diffe given general eqr be (03 +54 AD 1479 = 0 eman be a trial son of the eam in + m + 4m + 4 = 0 A.E. m 3 smo bm+1) +4(mar) @ 09, fm+19 (m + 1) = 0 m =- 1 g t ei general 2017 y = ce Sinzon ox. evi + C Cosene C3 Gigagle are constants دا - , - هلا y=0 when n = 0 - 2 Qq Bin2a + 2 C3 C08 22 Y = =, when on = 0 => -1 = - C, + 263 -cen e,=20g = 1 2 y = cea - AC₂ Corra- C sin 20. Y" = =.5 when n = 0 => 5 = C, AC By ④ f. 2 By ① & ② - 5C2 = 5 => C = -1 e2 + 2 = -1

2 Cz - 1 C2 = 0 = 7C3=0 ( el general 60th be y = -x +4 Co82 +2b at2 Let A.E., m² 2bm given diffen eqr. be dx dale dz tkon=0 kybo x=emt be a trial son of the equ m² & 2bm + kr 0 m =- 26 I 16² _ 462 2. = - b ± √b² -k? = - b ± √ - (K²_62). = -b tia where a = K² b ² yo general solo é bt(e, cosat+Cabinat) don Sna where a²= K²b at doa dt e-bt e-bt (-ae, sinat + aegcos at - be, cosat - beqsinat) { cosat (qa - a,b) - sinad lep tahl where a = R2-6yo. => 0 = 0,=> x = Cge - bt da = r when t = 0 => V = 'cga on=0 when t = 0 =4 sinat le Bo vt.in