Question

Solve the IVP -5 3 [= il (x1+[ 6te *) (O)=(-1) -3

a.
b.

Answer 1

[ ]]= [5 ] []+[t] [ce]-[ Let, A= Was find eigenvalue and eigenvector of A. 14-d1120 1-5-2 3 > (-5-2) (1-4) +9=0 +42 +5+=0 - 2 +42 +420 > (+27=0 > a = -2,-2 eigenvalue =-2. for a = -2 A-(-2) I = 2 2013 ] sikk. Las 2 S6 [: :-[ ][7]+[:) 2-Y 20→ney one linearly independent eigenvector, There is only 10,=1]

Now solve for n, where n such that, [; })(09:11 -31; +3^2=1. ri ☆ +42 1-13 +22 ::7 22 choose, 22-01 we get n = : [1] = «[1]*** 7.6:([!]4e24[:]<) ** D,-) (3) S - 2A ke 2+ + 3A ke-2t_2B the 2k + 2B tert [-2D t e 2 + 30 t ezt - 26 t e 2+ + 2E tert [-2At ezt +(3A-2B) 6€ 27 L-20 t ezt (30-2E) (?? -zt +2bt ezt +2Eteze + 6tet P-2 At?e=2&+(3A-2B)*"6-24 +2376-26 -20 t e 2t + (30-26) t e 2+ + 2 E t ezt [91. - [5 :][?]+[***] 1-ESTE Até ?*+B & e 24 Dd² e 2t + E t e 2t + F 1:14:17 1:16:35 (-2Ax² e 2 + + (3A-28) t e 24 +2 Brezt TOSA +30) te=24+(-58 +3£)te-24 +(-5e +36) L-200 Ezt + (30-2€) te-28+zEtelt |(-3A +0) t?e2* +(-38 +€) te=2*+(-3e+P) FC8A-3024e-2+ +(JA+3B-)#"ezt +2B teszt-(-52 +3F) (34-39)&24 +(88+80-387 +224 +26 terzt-(-36+r) Coteza 1-1073-[***]

..2A-30 =0 OF JE-887VE 2 ESO 2B = 6 > B = 3 2A-D=0 OFFE- st 166 ECO A =D. > A = -3 > D-3 -5C +3Fco -32+ f = 0. fc0 (01.- [***] [])--[]<"--[[*** [+]")-(*) [mcd][ - 4[:]+[3] + [:)- » [4*"] = [+] G - 15 4 =1 & 4 = ce 2-2 2 C₂ = -6 + (37= t»[1]**--([!]«c* [*]***) - [er] Peltsielt] -[2- [**") ezt a 6 teht e-2t_bt ezt +2 ezt -3x3 p.27 - 3t3e-21

b) (%) =(-1] [2] = [:* ?][0][*] let, A= [31] Now find eigenvalue and eigenvector of A. I A-Il0 zali -3 (4-1)(1-2) +6=0 3 ² +3a-4+620 => 1? +31+2O > a = -2,-1 > (2+2)(+1) =0 di = -2, dz=-1 eigenvalues are For, a = -2 4-6492-122 ] iba[5] [3] 1:10: (51 = [1] =»[!] eigenvector corresponding eigenvalue to = -2 - 1 for, de=-1 A-(-1) Ia :] SK. [ * ] R3*26*R[] :[][]-[:] 2 y = 0 ana 72 25 y

: []= [:"]=»[] eigenvector corresponding eigenvalue dq=-1&s V22 2 ܙܝ - + - 19).=643*[] tee*[3] Let, At to ct to [3]. - ( 4 -] = -1)] "CE] CH [16] -3 » [2] = [-3 1] [2* *0]+[5] » [A] = [ematik (960] + [6] ICA-20) ** (10-20 7690 10-20] - [6] "A-c=0 0-8-27 84 A +4B-2D 20 0 8-st set :. 9 A-26=-6 2A-C5-3 C3A 0-02-8b+6¢ 8=82-8bc 6 -=- IEC >>C=9 > 2 A-3A =-3 45 21 --83 2 -A=-3 A-3 3t [], 21 2 45 2 st

2 [2] [2] - eigenvector corresponding eigenualue dz=r2 is .. The general solution x-Itae [3] - set[] tie [1] + [3] YC) AC [1] [][] [1] 임 2 99 G 2우4름을 그을 의 t= 끌 G=릴 26 와 [3] - [1] + [ [ 2 e2t4ect t3k 릴 tbettek งไง งง 21