Question

1 of the questions remains unanswered. (1 point) Consider the linear system -3-1 a. Find the eigenvalues and eigenvectors for the coefficient matrix -3+1 AI 01 5 and Az 02 5 b. Find the real-valued solution to the initial value problem { - 3y - 2 5y1 +32 (0) - 10, (0) --15. Use as the independent variable in your answers. m (0) (0) Note: You can earn partial credit on this problem Preview My Answers Submit Answers Your score was recorded. Your score was successfully sent to the LMS You have attempted this problem 1 time. You received a score of 50% for this attempt e a 00 DO

Answer 1

We consider the linear system 1-3 - 2 5 3 :9 = Ag, where A- 7= CS-3] 3 which is the coefficient matix 5 3 a. The characteristic equation of A is det (A-a 12)-0 1-3-2 5 > (3-4) (3-7)+1020 –(3+2)(3-) +1000. > Bta) (3-1) -10 =0 → 9-1-10 =0. > aral :aati, i=h : The eigen values of A are air i and a2 = -1 Here az is the complex conjugate offi. Let vi = [] be the eigen vectors of a comsponding to the [-3-l – 2 → 63-i)x – 2y =0 and 5x + (3-1)y=0. 5x = -(3-1) y. n = -1 (3-1) y . y= S, SER. 1.X=-5 (3-1)s 5 → N = Let - Cj: ["}(0.93 au-1950) [***] 3-5.

5 cos (t) + where ů (t)= [ Ý (0) --3008 (0) - Sin (9 +i (CO3 (7) 3 sh (t) i5sm (6) (-34s (1) sin(t) Cos (t)-3 sm (t) 5 cos(t) 5 Sin(t) ů (1 + i uz (t) - Cos (t) - Sint 5 Cos(t) ů (t) I cos (t)-3 sin(t) 5 sin(t) Therefore the general solution of the system is gts et (+GUZO, 4 and E being, arbihang 1-3 cos (t) - Sin (1) -4[ 14- Sino + reos (t) - 3 Sii(t) 5 Cus (t) + C since y 6 -IS] oso 5 Simlt) We ware + 16 -15 ] 5G 5 1-39 1-3 G + 6 59 : -3G+ C = 10 and 542-15 → C = 10+ 3G. ►9--3. 10+36-3) =10-9 :. G=1

from equation ®, the have ..9 (t) = -3 3 cos (1) - Sin(t) 5 cos (t) -> | Cos(t)-3 sm (t) 5 sim (6) 1569] + " 9]+[ → 90 I cos (1) + sin(t) Cos (t) -3 sin(t) - 15 cos (t) 5 sin(t) g cos (t)+3. - 15 cos () + 5 Sin (t) Ilocos (t) L-15 cos (t) +5 sin(t) :9|lt) 10 cos (t) -15605 (1)+55mii (t) 92(+) : 9 (92 10 COS (1) 21) = -1500s (t) +5 Sig(t) These are the required solutions of the given initial value problem.