Question

(1 point) Solve the system 6 -2 dc dt 20 -6 C -3 with r(0) = -2 Give your solution in real form. 21 = 3cos(21)+8sin(2t) C2= 1. Describe the An ellipse with counterclockwise orientation trajectory

Answer 1

soln :- Ginen system 6 du at with a 100= -3 20 -6 > -2 Here the coefficient matrix is 6 - 2 20 6 The characteristic equation is x - tv (A) dt def(A) =0, Here tr(A) = 676 or da d. det (At = -36 + yo 122 니 Let v VE bany be the eigen vector (A-2) v corresponeling d=2i Then -6-20 (6-2) 21-222 (6+2)/2 20 u

from system (3-2) e-d2 = 0 10 e - la = (3-) lila o 3+ Horce one reduced system is (32) e-le=0 7 (3) de = b Let l = 3-1, (3-i) li = ds =(3-2) e = 1 사 Let V₂ = be the eigen vector correspording Then (A+) V2 = 0 6 trê 2 :) zo .6tri (6+2) oy - 2 2 ni 20 nt to tri) m = 0 from system (3 +i)n-m -O n - ₂ + 10 3-i - (3+i) ni ng 20 Hence om neduced system is (3 tin, - ₂ = 0

Let mg =(3+1), then from (3 +i) n = Nq = (3+1) ini 1 = NB 3ti Herce one required genical enical solution of the given equat nation tion is a (t) = Gu, é dat + G V₂ constants X(t) - G th e G, į are 2 zit 3 e (1) by the initial condition & (0) = (22] -zit e n(o) = 6) put to in 31 th -3 3 ti -2 ry 4 tlz =) G=-3-6 34 - iG + 3 C tie=-2

8 get putting the value of a in (3-1) (-3-6) + (3+1) CZ 5-2 om - 9+3ite 2 i-3 +3 ti] = -2 a [zi -2 tabi 7-3i С 7-3; 7+3 zi - 2 G = -3+ tit3 -6t7it3 2 - 2 -3+7i 2 Then it 71+3 - 3+7 Z ( -rit о urta e 3- 2 3 ti

-3+H Now XU = 3+7 2 2 os et tisinat) + (694 - isinu) (1+121) -tili 1-3 ost -7 - - Cont Sint 2 -24 24 (t) -3 art & E7) sint H (t) = (3) since - 2 coat K₂ (+) - 24 sinal Hence the required solution of the given clanation is e = - 3 coszt - 7 sinat (In real form) Rz - 2 szt - 24 Sinat Ans