Question

QUESTION 18 The 2 x 2 matrix A= has complex eigenvalues r = -2£i. An eigenvector corresponding 2 to r = -2+ i is 1-) The system x' = Ax+ (). e-2t has one solution given by x(t) = (2) e-2t. What is the general solution to the system? C1 cost sint - cost 6) -27 +02 e sint sint - cost :) -21 e + (1) e OB C1 cost cost 0-2t + C2 0 sint e-2t + (1) e-21 C1 el-2+i)t + C2 ( 1) e24 (-1-3) (---:) 1 OD Ci el-2+i)t + C2 -1 + el-2-i)t + (9) e-21 OCI cost cost e-2t + c2 0 sint -21 + C3 e-2

Answer 1

The answer is option (A).

Given : A= has complex eigen values 2. JT 1=-2 + Am eigen vector E = (1 i corresponding to eigen value r = -2+ i The system x' = AX + -lt has a soln برم (3) x64) = (3) ezt st 3 e To find :- General solm of system sc'> Asc + Solm. We first write the general soln for x'=Ax the eigen values are complex, we can get 2 linearly independent solm using an eigen value and corresponding eigen vector : 0 = = ) A= constant x A erit (-2+i)t 11 A e (1) (4) it A e 2t е A e 2t 10 (cost + i sim t) = A e 2t cos t + i simt -cas it & sim t - i sim t-icost) CS Scanried wi CamScanner

-2t => x=0, e cos t -20 & C₂ e sim t l-cos t + simt -sint-cost 'We separate the term with with ü" and take it as an independent solution, -2t X (A) (3) is a sclution to I': -2t AX + 3 General sol for sc'' (3 jest + As =) -2t се = + Gez X + cost simt sim t-cost sim t-cost) - t + () e ans. CS Scanned with CamScanner