Question

Please Solve Both Problems

Clearly and neatly show all work for each problem. Solutions with no w credit. 1. Solve the following initial-value problem. X'=(123), x(0) = (4) 2. Solve the following initial-value problem. X'=(-2 ->)x, X(0) =

Answer 1

SOLUTIONS x'= (2²) x x(0)=() x = AX Eigen value equation for coefficient manix 1 A-Il=0 U 3 6-1 I -2 (1-x) (6-1) +6=0 d271 +12=0 (2-3) (4-4)=0 d=3tda = 4 Eigen vector corresponding t,=3 (13&][%]=[0] -2 3. 2 . = > [:] -2 3 R₂ R2-RI > (2 :][*]-[:] - 2x + 3 X2 = 0 here x is free variable choose x₂=1 => 2 = +372 3/2 U= [ 0 ]

Eigen vector corresponding t2 - 4 1-4 6-9 -3 S 2][3]-[8] [ ][:$]+[:] -x+ X2 =0 X=1 to T2=1 7 V₂ = [1] Hence solution. x= cient cient u + Ce etzt V2 X(t)=G given X (0) = Gehe [**]+22"[] [4] [4] = 6 (14)+c.[:] [4]= " 3/9 + C₂ C+C2 G + C = 4 = 4-6 + 3/29 +22=1 / 4 + 4-9=1 #8718/2 After solving (G=-61 (6 =10) Hence comple solution (x (+) = -6 83t [ 3/2] the pet [!] SAE

SOLUTION 8 X' (t)= x/0) (2) Eigen value equation for coefficient mabi's -2 (1-2) (-1-1)+2=0 - ²-1+2=0 2+1=0 d=ti Eigen vector corresponding d, = i Hi] [.= [8] R2 Rat (ti)R, - -2 X- [ O] (3:10] (1-i) x + X2 = 0 Choose 22 - 2 Vys - 2 as free variable - Citi)2 - =-1-1 Hence eigen vector 2

Hence x Ct) = it [-] = Ccost ti sint) 2 - cosat +Sint +i (-sint-cast) [ *- casts] 2 Cost tdi sint sint -cost B, C+) -sint -cost [ 7 04] 2 cost 2 sint [-sint -cost Hence complete solution X(t)= G sint-cost 2 cost + C₂ [ a sint Now [ sint ost] x(0) = (2) (?) 677)+10) given -) ( -G-C2 G=1 4 = -4 4 sint + Host tence -8 simt * 4) = (sintereses + X(t) = [. cost [-2] [3] + sint [5] 5 sint - 3 cost - 8sint + 2 cost ] OY X (t) =