Question

#7 (50 pts) Use an appropriate method to solve the given system 1 8 X'= +

Answer 1

T9 102? 'het Ä 8 A = - - ľ -O -- ht to the Then AK the system be x'- Axt fett, where fit= let Characteristic ear of a in 8 It =) trg=0 =) d=13 k" = (4 be an eigenector corresponding eigenvalue DIE 3 dik kell given -2m +8K2=0 ki -4K L=0 equivalent system is ki 4 K₂=0 ky (Gay) =) 4 = 4C, K2=c (CEIR-30%) c=1 kel=14 Similarly eigenrector correon ponding the eigenvalue the is ht K4 -9 (4) tą est/2) or, X casę are a.l. こ -). = choosing to k(2 = (?) (۱) edit Jo, Xn k te é 3t e

3t Fundamental matris in 4 e p(t) -ze-3t e 34 e-3t e-3t adi [olts] =( e 3t -3t -ti ze he 13t deptolt 4e3t 2e-3t 4 e3t det [pley) = 6 [pits] ad; [olto Then [pits] fen é 3t ze ért 4o3t -2t tité e2t + 4t e -36 be Corey tet, et ut e ut -zt te £e-2t. dt= at -te ut (- 1 e 2tt teht to Integrating J [pt] fet, da tte ut teht particular solution in Xp (7) = olti / [pl] fits.de 4e3t - 2e-3t - 20 ²t / tette 24 helt e-3t l-tetrtett - eft ut et rett fatetttet -tet yet het £et yet t =to (h at 3t e teht seht e G C et t

So, general Jorn in X=Xn (A typity =( - Zet_btet 6 24 et 3 het Or, x= Ge (4)+422=3+ (?) il - 3 e tbt et + -3 3 et 4 et - 24 (req.)