Question

Consider the following linear system. = -x +y + 6z x+y-Z = y + 5z a. Find the characteristic equation of the system b. Find all the eigenvalues of the system C. Find all the fundamental solutions of the system d. Find the general solution of the system

Answer 1

ear em oux :- consider the following linear system dx = -x +y +62 © dy x+y-2 da = y + 5z @ Find the characteristic equation of the system o find all the eigenvalues of the system fund all the fundamental solutions of the system @ Find the general solution of the system solution given linear system du = x+y+62 2. de ex+yz @ du - - - dt xty-2 equation can be renerite us - 1-1 1 6 11 u ZJ Lo iste I-111 where A= ollo 15 . Ti 11 is officient matrix of A find characteristic equation of the Ryslem Characteristis equations det (A-1) = 0 arral 161 lloof II - - 01 020 To is lool - 1.169 11-1-00 ois loon od 0 11

-1- 1 6 alia - 1 | 0 | S T ) |--2 16 I 1-4-1 © 9 (1-2) I (1-) (5-2) 4) - 11(5-)] +6x1=0 (-x) (5 -> -534+x+1] -$+*+6 zo = (-1-x) [x² - 6x +6] + x + 1 =0 = x² +62-6-x²+64² - 6x +x+) zo 1-23 +50² +5u - 5=0 w 5x² - x +5=0 Characteristiè equation Eigenvalues root of characterisie equatim is eegenvalues x 3 5 x ² x +5=0 x²(x - 5)-1 (x-5) = 0 (-s) (x²-1) = 0 (x-5) (x² - 1²) = 0 (25) (x+1)(x +) = 0 x=5,-1,1 fance Eigenvalues 14=5 , 12=-1, 13=1 | so

o fundamental lest Yi=elt Y = est sets of solution: , Yezetet feest ,q=et, gazet 3 fundamental sets of solution & W (Y, Zifs) to Wly, z, y) = l est et et I sest et et lasest et et - est l-ette ezt] - eelsest et efwere tet l sest et + sest et est 1-1-e²4J-ee Leot 25 et ] + et (seet +25ett) -est, et est +25est the 7t + 25e4t c Wly, if, f) = 4 ett tasert +25e42 to lest, et, eff are fundamental set of est General solution x= cielo, & caetet e f Cz e3f Us of 1, 2, 3 .. where U,, U2, Us are eigenvector for 11=5. Hence @ are distinct

LA-Ile=0 161 II - Solo oo | 015 911 A2 = 1 OOO Pall 61 15oor ( 113) ( H ) H18 6:18 Lois loos -1-5 I 6 1 A2 =1 lo I 55 L -6 16 1 Paino 11-4-1 Az 1-6 1 61 6 -24 -6 1 0 1 0 23 a21 3) 8-18 R3 23 R₂ + R25 A 2 P-6 1 61 To-23 o Loool - 69, +azt 60g = 0 - 23 a 2 = 0 + 0 =07 - 691 tot baz=0 - 691= 6 as 19,=a3] Hence 0,= 10

for 2=-1 (A-1 U2=0 - 161 Ⓡ1 - o 15 lloon 0 4 :18:13) 02 lo.o a 161 loog) 1-1 + o 15 Loo coo Intl 1695 I 11- lo I 5411 JE 1801 so 1 I 2 6 - 1 ooo 2 so- A2 = -693 a at 6a3 = 0 - a+202 - 03=0 a+2x - 69₂-93=0 21-1293-9320 9 - 1393=0 A1 = 1393 mann Hence 13 1-6 U2= How for 13=17 (A-1f) U3=0

- 1 6 1 007) 1--1010 18+) 2- @ o 1.5 Lool 1-1-1 162 lain OL 1-1 -1 Loi 5-1 1-2 1 6 1 sal 1 O -11 o 14 J L 93 1 1 -2 1 6 1 an 2o-2-11 to 14 JL az -) 8 ) - foto 8:18 fotot 1-2 161891 01 41 22= Ritka e Io 14 Las, 1-2 1 61 {Rs > R3-R2 C о с LoooL azu -291 +9₂ +693 = 0 a2 +4 93=0 192= -403 - 29 - 403 +693=0 -290+2.93=0 91=a3]

Hence " @ General solution X= Gelet uitleetzt ut geht uz x= Cret * T 13 1 -6 1 *-42184414144-14] 4 1 1 + (z e -4 Ix- ciest + 13 Get teset I -6Getaget ciest + Get tet solution