Question

Linear Algebra

system of differential equation and symmetric matrices

please elaborate every step so that it gets easier to understand

thank you

6.3-Systems of Diff Eq: Problem 1 Previous Problem Problem List Next Problem (1 point) Let (t) = be a solution to the system of differential equations: 22(t) (t) x'(t) = = 331(t) + 2x2(t) -11(t) If x(0) = , find r(t) Put the eigenvalues in ascending order when you enter 2(t), 22(t) below. 31(t) = expl t)+ exp 22(t) = exp( t)+ expl Note: You can earn partial credit on this problem. Preview My Answers Submit Answers

Answer 1

Solution Ceinen x(t) = xo(t) ~t) 30 (t) & 202 (t) xat t'k- Can be written in The Grive on Matrix to Form /Bro(t) of 222 (+) dr. - 'k- ( 3n, (t) + 2x2 (t) E = dt - to Differentiating (1) warto 't we have have op 3 de nos (x-) -ampen o = (4) k22 TONPE- 7

if y = x (t) - 3 +27 - (Hy Now Now let Ya emt (where om being constand) te a solution of the coresponding homogenious Equation of (ii) Then ifs auxiliary equation IS om - 3m + 2 = 0 > int - 2m - m + 2 = 0 m(m-2) - 1 (M-2) = 0 m=1,2 ..yo ciet e le e2t. axi (t) - Ciet ec e 2t where CC are arbitarry constants Now 2 + dat d ㅏ ciet e 2 cael JX It Cat 24 down () 222 (t) = d - 3x, (t). - Ciet ecze - 3riet - 32 e 2" 2-2ciet - Cze 2t "2 (t) = ₃ (-2ciet - lze2t) - - 20e

2 Now wx (0) = ² (-29 - (2) x. (0) = Circa 6:03 - 2 C 1 and C, C = -2 Ciel2 = -5 and and C2 =-5-C1 ::-2, - ž ( -5- 6) = -2 - - C, + 4 + 2 a) C = 9 - (2=-5-9 = -14 : xi(t) = q et alueet x2 (t) = + (-18 e + +14 e 2t)

x (t) = get -14e2t Ź (-18et flue2t) 9 eup (1) te flu) enb (2) t' (t. 1 = 2 (t) = 9 enp (1) t e (7) enk (2) t