Question

Use the variation of parameters formula to find a general solution of the system x'(t) = Ax(t) + f(t), where A and f(t) are given. 12 18 - 6 A= f(t) = 3 6 - 2 12 X(t)

Answer 1

hiven: -6 A = t-2 -18 6. -] f(t) = 3 -2 t-2 Eigenvalues of A are (AI) 118-1 - 6 = (-16)=0 -2-) I=0,16 Eigenvertor for d=0 is [A-oI] =0 18 -O 103 -) 6261-222 = 0 -2 Let us take x=1 we get (x2=3 -> V= eot pilt)= eft I - = ) ... 4.(t) = © Eigenvertor for t = 16 is [A-16]] v2 = 0 201 = 0 = 2x1-6x2=0 -18 let take 1x2=1 we get 2 = 3

Va (?) = 42(t) = e = e lot (3) ::0 (+) = [0, 1t) falt)] = $(t) = 3e 16t el6t 3 6-"(t) = -1 e 16t e 16t 16t -Be 8 &-' (t)= 8 لما من 3 e 16t 8 -let 2-16t 8 • '(t) f(t) = t-2 3 s 6-'(t)f(t)dt = 3t Particular integral Xp = $(t) S *-'14) f(t) dt 1 Xp = 3t 3 e 16t e 16t 3

Xp= 3t in ito t .: The general solution is a(t) = (1 $.lt) + ca dalt) + Xp =ci (3) + (a elst (3) t 3t t (1+3lze 16t (t) = 3t 36, +(2 e 16t - +