Question

please answer asap thank you

dx / 3. Solve the initial value problem: live the initial value problem: os - ( 2 )x, x(0) = ( 3.). To dt

Answer 1

consider the given Ivp x'= AX with Xco) = [2] The characteristic equation of Ais 1A-XI)=0 ~ 4-1 1-0 1-2 2-X (4-x)(2-x) - (-2) (1) = 0 X-6 +10 =0 x=3 ti The eigen values of A are x=3-1, 3ti consider the system (A-XI) X=0 cohen = 3+i we have (A-(3+i)I)x=0 1-2-i-i Laa Lol From ist row: (1-0)# +22=0 » X2 = (2-1), The eigen vector ve when x1=3 ti is

The solution to the eigen scalue x1= 3+1 and the corresponding eigen vector viole is 7 X (t) = 3 (cost trantsiz. 3t [cost + i sint Licost-sint-cost-isint] t = 3t cost + 3t sint 7 Lest-sint] The general solution is -sint-Cost .c, 2[c.com] celor use 20-61] - ()--[]+48:] La+C2 By equating C, = 2 and C2 = 1 The solution of the given I up is X(t) = [203t cost test sint L-2etsint -2ét cøttet cet est sint] ret (2 cost + sint) L-e3t (3 sint + cost) (t) =