Question

Use the method of variation of parameters to find the general solution of the system

x' = [2 21]x+[287] Ax + g(t)

Find the Laplace transform

Answer 1

Given x = [e &] x + ( 2ct - Ax+ g(t) Variation of parameters - for general solution XG=XH*Xp - for homogeneous solution, get)=0 x= AX- 2 C 2. find eigen volves & eigen vectors of A (A-all eigen varies =) 2 2. - 1-2 (61-2) (-1-2) - 450 1+22+2²_450 -) a²+2a-3=0 a = -2 ± 54+12 -2 756 - 2 3 4 1 -3 2 x = -3, 72 = 1. eigen vectors are (A-2) V=0 = x,v & Az - V2 (-1+3 2 X2 2 -1+3 2 2x+2x2=0 = x,+X2=0 (2 30:23:13 x = -x2 take X2=1 2 X=-1 So for 2,=-3, Vi=

for 2 2 = 1 2 -- -] A-C M-a ( 2x3 = [] 2,=82 - 2x, +2 Xe=0 - 2 2x - 2Xq=0 X = X2 take x2=1 21=-3 - 2 = 1 vp = { homogeneous solution is XH= cuerit + C2 eart XH = C -3t + C2 -3t XH = C + for particular solution ay per variation of parametely xp (t)= X(t) x '(t) g(t) It Ixe where X(t) = (x, xq] fore forom XH (t) we got 9, x, +C2 x2 = Г-e3+ + C2 e-3t é 3t X(t)= 윤 [ X'(t) = 1 - 24 -2t -e -e -34 - 2e2t - -e3t -e3t La x"(t)= (a est e3t

xp (t) = X(t) X(t) & (t) et cest et - est e3t lt et ét 3t - e 3t ot -2e²t + 3te3t dt 2 e3t et 2e2t +3tét -est ezt + 3t 9 ca 3 (test é 37 et + 3 (-tet-et) -e-3t - ezt + + te3t - est ezt a tet- 30 e-3t et Bet et (-4t-8/3 == -t + / - et 3t-3 Letttag- ét_3t-3 - ام L-rét_2t - 10 xp (t) = --21-4/3 -ět _t-5/3 XH + Xp - e-3t -et-24/3 cu + c2 + e-3t -et- t - 53 XH Хр is the solution.

we know that 6 Given L ( sct-1) cest) [ f(t) SCt-a) = 0 itt ta Jef (t) s(t-a) = f(a) it t= a » Caplace toans form of f (+) {Ct-a) est f (t) & (t-a) at e as f(a). ...Cast 8(+-)) - ES (1) L ( f (+) 8Ct-a)= e as f(a) hele f (t) = cost s(t-a) s(t-1) a=1 :.. (ast s(t-1)=éscs (1)