Question

1 ]x 3. Solve the initial value problem. 0 0 1 (1) x = 0 10 X, X(0) 1_0_0 5 =3 0 -4 (2) x = -1 -1 -1 X, X(0)= 0 1 1= 13 = la ;1- =ܬ [ 3 2 0 lu-la = lg=-1 1 1

Answer 1

3.00 t": ♡ 2 X oJx 1 X101 - 2,:-1, 22:13=1. 5 Let (24, 24, sy]t be an eigen rector forrespon corresponding to the eigen vakre at Then 10 otell 10) - 0 0 ol - M -7 M₂ O 2 o N 0 4 tlz co 211220 Take dip thandant Take Az = -1, then y=1 eigen vector, - Thne : ] is am 1 Let [\,,nong]be an eigen eigen vector con corresponding eigen vahe to the 1 0 g. - Then, 0 1 0 " 16 0 0 0 0 , 27 "2 Ч» M, &Y 20, Y,- 1320, lett, arty Then eigen reefor ty (1):19] :) are Thus eigen rectors com responding to I T:1:10] -

Thus the soln is MILE oglajene +6,18% Getthet get -Get tet Mow, X1097 Get le 1 2 >> 2 5 - G+C2 - G+q=1, 302, -4+2=5 -X (6+2)+(-4 th) = 115, (322 = 3, C=2 27 ₂ =3, (3=2, G = -2. :. xlt) = (:) [:] > ₂ - 2 ért 0 + 3 et + 3 e 2 et +30 zet 2 et +Bet 3 0 XP : :)*, xoli 12,=12:23 1 =-). let [no da ray]t be an eigen ree for corresponding to the eigen rahue -1.

Then. (6:6:98-10) - -2 0-41 24 72 -1 -1 dz 2 1 2 - =7 - 2x4 -4 03 20, - 24 - 2z20, 24+ 2 H₂ 20 -2 4 = Uz 1 = 0. Thus eigen weeter is (:). - lety [", " rectors that satisfies นา" and [2, 2₂ Zz] z be two A+I) y = [:) ano (A+1 )2 =Y Now, 4 1:33:1:) " I - =7-29, -4% =0, -", - Y₃21, 4, +2%=0 => 9,+ 2 Y₂ =0, -y, -73=1. Yg 4, 7, = -2.

:-y17] 2 o al Again 2 22 - 2 Zz =7 - 22-423 = -2, - 2 - 23 = 1, 3+223 = 1 sy 2 +22 + 2 Zz=1, -Z+Zz cl 37 ? =2, 3 = -2-42-3, iriz 3 2 :-X1t) - G le + con (ili T:J-17 Denet (*[]-+77)+(374% ta -26ert c zoztert -36 ent + Gtet Caert +la ett + Cg tert Get +Cg tert+lztett get +2 Gent [ 31-> [-24 -365 3 2 D Xco- G+62 +63 G+ 26₂ 27 -26-363 = 3, GtqtC3=2,62 +263=0, >> C2 = -213, C3 = 3, G+(2+(z =2 =7 C =-6, G 2276-3=5; (3=3. in Alt) = 120-tabte taget -bet_ftet aset+3f²ertaste-ttet -bet tolet abert sent obtent zet-3tert +8t²ert 3tét
I don't think there is anything wrong in the calculations. I have checked it many times but don't forget to check the calculation. Please let me know if there is any calculation mistake.