Question

Solve using linear Differential equations

1 -1 1 1. X'= 0 1 3 X 4 3 1

Answer 1

2=52644) 21261). aucto Solution :- for a system of fisst andler defferential equations. ' AX, LE* Suppose that A has in eigen Values qindr, du with Congespuding elga Veeters V, V2, ..., Vn Then The general Solution to (4) is is given by 2 - nict) H266) Gent V, + 2 / 2 eht t +Cr Yneant : XnGt) -1 0 1 3 43 Here n=3, A= No tokresponding characterisht Cerertim is Tatoo - det (01 –A)=0 - 26-1 det for N-1 -3 -4 -3 2-1 = 633212 - 100 + 24 =0 - (2-1) (94—2) (22+3) = 0 =) eigen Values are 1, = 4, 12=2, 23=-3 fivel-eigen-Vectors, Alext,

11 = 4 Consicles the System (A=9, IX=0 sou nt 2- 263 E3-1 24 0-33 le 43-3 913 -571-77127213-0-YO - 3212+3713-0 All +372-326370 6 > N2=113 -311,0 => 711=0 =>7. (). (1)-(0) The Spure of eigen veefoss of A Cossesponding to 2,=4 is given by Fuq )= {m(?): 1a EiR2=Span 2003 = An eigen Veeter Corresponding to do=4 Con be taken as X (1) 12=2 (A-21)X=0 24 -I- 0 4 3-1 -13 :) Xa 23

> 0 -N-N2 +213-0 -262 +326=0 404+5=113=0 2 1/2 = 5013 @=> = n4=-2013 -11-3763+73=0 1-2013 lou 12 H 262 3262 012 E = 1/₂ = Can be takey_as an eigen Veetes of A (esresponding to 12=2 Az = -5 A 2X=0 → FA Anu 3) o As 262 A 1213 쓰요 > 0 421 (2+83=0 4912+523-0 7491, +3712+47/3 -0 → 5 2 -=-5717 4 O=42+3213-423=0 可 ALI -7213 -7/16 n13 4X4 fot -71613 22 K3 3423 913 = = x2 X= (2 -7/6 -_913 hu Tol-12 16

=> // 1-7 Can be taken as-an-eigen Veetos of 4 COBASPaneling to 13z-5 Hence the general Solution to the System -1 os X IX is given by A3 L 37x X = (all (266+) W211) X360) aviat + 12 / 2e azt +63 Vzedzt At e 7lz Ez to Be -2 3 e at +63/42 lest 16 - Alit) = -262 eat_763 e 3t 12 (1) = Geht +36 edt + 12 C3 e 3t Geht+Chekt 16ce36 213(+) =