Question

(1 point) 21(t) Let X(t) = be a solution to the system of differential equations: 22(t) (t) x',(t) - 12x1() + 2 x2(t) -10 x1(0) 3 x2(t) If x(0) [:] find (t). Put the eigenvalues in ascending order when you enter xi(t), 22(t) below. * (t) = exp( t)+ expo t) 22(t) exp( t)+ exp( t)

Answer 1

given z(t) = -128, +222 xact) = -108,-312 can be written as x'= AX where AS -12 2 & X= x= 3 -10-31 the characteristic equation is 2 (-15)d+(56)=0 d²+15d +56=0 d+14+8d45600 d (147)+862+7) (d+7 (448)-0 d=-8, 7: are eigenvalus let di=-8 se d2=-7: eigen vector corresponding to do = -8 X -1248 2 -10 -3+8 -4 2 -10 5 R₂ Rals -4 2 2112)=[:] -2

R27 -2R24 R| [ ] [3]-1} -14 212 = 0 기 02 222 let at thens 72=al-i [st] Jet Vi= [5] be the eigen veelor corresponding lã dy=-8; veelor corresponding to d23-7 eigen X -12+1 -ID -37 -52 -10 4 128] Ay R21 R2-21) 15 El-e} -51, +27270 5% = 2기2 가름하 l와 5t 빼계와 차 파 t

let V2 = [ ] be the eigen vecter concesponding to dig = 7 The general solutionis X(t) = Civiele tega Veetzt (32= a[1]** G[3]et Gest262 e 76 2 21 (2 nr -86 260 or +562 52 et then ② becomes as 700) [ ]= [4] G+262 =1 26, +5(2=-1 201+462=2 261 +562 = 1 -(2=3 1 (2=-3 then (1=1-26=1-2(-3)=7 101=7 Then st_bê -76 Te che ha 7,64) 9261) 14885-15e %2Ct)= 7 exp(-8t)+(-6) exp#] 72(+) = 14 exp (-8+)+(-15) expl-17)