Question

Find the general solution x(t) of:

x'' + 4x = 3 cos(2t) + 4 cos(3t)

using the method of undetermined coefficients.

Answer 1

gall +436 =3 cogat+4.cos(3) Homogeneous equafies ise x 44 x 50 =Bxt 4250 10²4 x 50 put Dero, x=1 & m² y 30 m²=4 sms £ Ri X(t) = ACosat + Brinat

Let X(t) = A Gsin at +Gg coset ta sin3t+ - Cy cos 3 Zold) = 24 cosal - 26 linat +367 20131 - 3 Cylingt xp (t) = - 4G sinat - 46 Casst - 90 lint - 9 Sy Cox 3t Wring in en we get, - 4G singt- 46 Cosat - qadin3t - 26, 20836 + sinat tu C Lorat &Y G Sinat +49 Cosst 3 cosat + y coszt - G - 56 sinstasy Coost = 3 Cosget y cost - -56=0, -54 = 4 . - 630 = 625 Hence the general solutions into X(t) = x (x) + xp () = A cosat +Bsin at tG sinat to cosat - 4 conse = (At) Cosat +(Bta) sinat - Coszt Hence, a(t) = C cosat + D Sinat - 4 const where c&D are constants