# Homogeneous system

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Solve the linear homogeneous system...(see attached)

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

the characteristic polynomial of the given matrix is

(x - 1)(x + 3) + 5 = 0,

hence the eigenvalues are -1 +/- i = a +/- bi, where a = -1 and b = 1.

An eigenvector corresponding to -1 + i is [2 + i, 5], which has the form [x_1 + iy_1, x_2 + iy_2]

where x_1 = 2, y_1 = 1, x_2 = 5, and y_2 = 0.

Recall that the general solution has the form

e^{at}( k_1 [ x_1 cos (bt) - y_1 sin (bt); x_2 cos (bt) - y_2 sin (bt)]

+ k_2 [y_1 cos (bt) + x_1 sin (bt); y_2 cos(bt) + x_2 sin (bt)])

so the solution to our system of equations is

e^{-t} ( k_1 [ 2 cos t - sin t; 5 cos t] + k_2 [cos t + 2 sin t; 5 sin t])

where k_1, k_2 are arbitrary constants

Â© BrainMass Inc. brainmass.com October 4, 2022, 12:29 pm ad1c9bdddf>https://brainmass.com/math/linear-algebra/homogeneous-system-215814