Question

A system of two differential equations with real-valued coefficients has the complex solution pl-245)+(-3+2i) 3+2i) Then two real-valued solutions are given by -211-3 cos (5t)) a) - 3 cos (5t) and -212 sin (5) € 2 sin (5) b) -21(2 sin (5t) - 3 cos (5t)) 2 sin (5 t) + 3 cos (5 t)) and -21(2 cos (5 t) - 3 sin (5) (3 sin (5 t) + 2 cos (51) 21-2 i sin (5 it) - 3 cos (5 id)) 3 cos (5 it) - 2 i sin (5 it) and - 2 2 cos (5 it) - 3 i sin (5 it)) (3 i sin (5 it) + 2 cos (5 in) d) -21-2 sin (5 t) - 3 cos (51) 3 cos (5t) - 2 sin (5 t) 22cos (51) - 3 sin (51) 3 sin (5 t) + 2 cos (51)

Please solve these two MCQ questions.

Answer 1

(1) in solution 727511t complex form -3 +21 3 +21 1 ~3+21 3+2 i , tesit Art Los -e-2t (st) + i sin (st) ] 1-3f20 / 1 3+2i (-3 cos (st) +2 icos(st) - zisinist) - sinist) 3 Cos15t) + 2icosist) +32'sinist) . - 2 sinist) - 2t (-3cos(St) - asin(st) ti (2cos(st) - - 3 sinist) 3 Los (56) - 2 sin(st) +i (2001 (St)+35h ist) =utie =e-24 1-2 sin 15t) - 3C03 (57) | 3 cos (St 1 - 2 sin (st) and 2t / 2cos (5t) - 3sinist) | | 20s15t) + 3 sin (st) / option @.

R X'= 15 21 elgenveleus: 15-rd ? 1-4 -1-nl - - (5-2) (142) +B=0 de unt 320 a 2²-32-2+3=0 - (2-3) (Wil=07 221,3 and eigenvectors corresponding to eigenvalles? 2 = 3, v = (-7 general solution x-Gest 1 : 7 ] the t (22) given allo) = (?) -(, - £ ¢=9 Gt s t q = Its sad(9rs) x= -11815) e3t (geslet 0+3)e3+ +(88123)et / not possible for any s.

Note: to find the two real solution we need to simplify in the u+iv form as explained in the image