Question

(1 point) 21(t) Let z(t) be a solution to the system of differential equations: 22(t) = x(t) xy(t) -7x1(t) + 622(t) -8x1(t) + 722(t) If x(0) find z(t) 3 Put the eigenvalues in ascending order when you enter x1(t), xz(t) below. 21(t) = exp( t)+ exp t) 22(t) exp( t) + exp( t)

Answer 1

Given question solved .
(7) he 22 (t) let x (t) = be a soluation to the system of diffrential equation. x '(t) = – 724 (t) & shalt) y (t) = 824 (t) + 722(t) if x(0) - Find att 3 Put the eigen udues in acen reending order when you enter X(t) & (t) below xy (t) = esp 2) %, (4) = -l exp ( ) + 2] exp( ) x 6 = -7*(1 + 6x7 () day - 72 (t) + 6X2 (2) x₂ (t) - -84 (t + 72 (+) (1) ZŁ + () 128 dt XP — 7 dt J х 7 where X = 24 (1) La efferent matine is The lo 9 -7 f 7 let, w, -7 A = tra: sum of d'agonal elements - 8 7 -7+7=0 'N'be the eigenvalus of A and chanectenstie equation of Ais 4² trai x + deltA=0

and A= -7 6 > ² A. x + det A = 0 => 2-0.1-1=0 K 8 7 det A-– 49-(-48) => x2_1=0 - dett-– 49+48 -> (2-1) (+1= 0 det A=-1 7 ==0 1+1 -> x=1 x= -1 so, x = 1 - 1 Acending orden ualues are ا را - so, ut, Increasing ordene of eigen ualues 1 = -),, (acending ander) x= -1 and 4, is the eigen vector corresponding eigen uale A1 = -1 AV = V = AV = -1.4 (A+1) "=0 => [(19)+(: 011 m.) to the so -7+1 6 + o all -sto 7+1 - 6 ali -8 a21 o -Gaulthari - 8 allt 8a21 11:

--> - 6011 + 6 291 = 0 and - 8 a1 + 8 al = 0 all taal and - all + a2 = 0 au al so, all a) ( au azi am so eigen meton - (1) for x = -1 (0) and W, N2 is the to the eigen ualue A2 = 1 eigen weeten corresponding sol A 2 1.72 => (1 - 1) x2 = 0 => 012) A-I 910) (679)-(.*) a 22 X2= a12 1-7-1 6- 6 => 1 - 8 012 + 6922) fo 7-1 - 8012 + 6922 10 8912 + 6922 = 0 8912 = 6022 => 4012 = 3 a22 12 = ( and 4 40121 4a22 3 a22) 4 arz T M 3 4.

into a ☺ Now, substitute the value value of G and en et + 2le 24 (1) 21 et (t) است - +28et => 44 (t) = t + 21et H (+) – et + 28 et