Question

Find a general solution of the system x'(t)= Ax(t) for the given matrix A. - 6 10 AN -4 6 x(t) = (Use parentheses to clearly denote the argument of each function.)

Answer 1

OY; values are +21, -21, Let dia 22 and do = -21. 2,22 Then, (A-(2:11) e = 9 10 on line 10e27 The given system is &'(t) = AX (t) where A= where A6 The characteristic equation of the matrix. A is det(A-21)=0 - OY; -(a+b)(-a) +40 =0 or; (6) + 40 =0 or; d-36 +60=0 on 2+4=0 ia = +21 so the matrin A has a pair of complex conjugate eigen Let e = ( be the ergen-vector corresponding to the eigen value - (6+2) e, + 10 e2=0 -40, +(6-2) = 0 - (3+ile +56=0 , -29, +(3-1) = 0 A non- trivial solution of this system is = 3-1 = 2. ( getice; where - fil]; - [] Again, let go be the eigen.vecten corresponding to value Agro Zio Them; (A-(-231) - 0 -6+21 -4 11-6+21)9, +1092 OT; -G9, + (6 + 2) 22 = (-6+21) 9, + 1092 =0 -49, +(6+21) 92 = 0 OY, N 6-21 a Then; e = the eigen Or, 6421 ro V

A non.lrivial solution of this system is 9,=3 ti, 9 = 2. =)-(+34)9,+592 = 0, -29, +(3+1) 9= 0. ait ПЈою; and $2 a imaginary part (2 ) Casat - 9 = ( (suit] - []+[:] &,tige, where = {}]... - [] Contidele? +(2+2).(C0926+ i-Sinek) = (C082- Sin2t) +i(... 3in2k+S, C082) - Then; Si = real part of o [3]cosat - [j] Sinet of [3] Sinzł + Hence, the general solution of the system is 2(t) = 48 +9 (whene, que arhitung. e) =¢{[3]co [j] sinet] +6 {[] 31,26 + [2] szt] Coszt - (] Sin2t + (3] Sinet + (2 (39+4 20+ 2G 5 (39+es) co224 + (36-4) sinet - X(t) = @4+24) C0324 + (20-20) 31122 Caset Cosat Coszt 29 262 = Copat & (3₂-97 ginal 20-29 J