Question

'= [_! :]ā, co= (23) a. Find the eigenvalue ), an eigenvector v , and a generalized eigenvector ū2 for the coefficient matrix of this linear system. l= , Vj = b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in you answers. y(t) = 41 c. Solve the original initial value problem. yı(t) = yz(t) =

Answer 1

Sol" (a) consider the system of differential equation I then A-XI = 17 17 ao 5 => A-23 = 175-27 so def (A-XI)=0 » 127 5x1=0 -) (7-4) 65-1) +=0 - 35-79-5)+12+1=0 > 12-191+36=0 = (1-6)=0 so for ergen value as x=6 . x=6, we get A- 1 = A-67 7- 6 -1 5-6 performing gow operation, we get Riga Path Too on solving (A-61) V=0; we get [ ]N = [8] =) 3) Vp + r = 0 Vi = -V2.

let Up=t, then wi=-* Hence V - Fix] so one gen vertor associated with a=6 is V = EJ = 77 for solve second ergen vector, we need to (A–67) v = vi => [-]N-7] - 617) 3) Vi tvz = -1 =) YE--V2. -up let v co then ni=4 so V2 = 66

Hence, we get 1= 15), V = ), B1 By now generare solution is Jau = cien vit cq (tele v telt Va) =4,264 [i] + < (82 69 +€5* EJ) OOR te of using initial condition po) = Fos= FI] we are 13]. weget 3101 = 4 [i]+(? [5] - [U] 3) -,-17= (p= -6,+1= ~2+1=-/ substitute (o 1,67=7 in PC41, we gee get = (-) het J- Live on J - (Re***062 +2647 JL-2064 6061 / Therefore Vict) = Ja (41 = 3064 +48 64) -266f_ pe 68 CANS)