Find the general solution of x'=(2, 1, -5, -2)x+(-cos(t), sin(t)).

(this is 2x2 matrix with 2 and 1 on the left, -5 and -2 on the right. And -cos(t) on top, sin(t) on bottom. I've found the answer for the 2x2 matrix but have trouble finding it for the other part. I want to solve it using undetermined coefficients but I don't know what should the form be.

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You found the homogeneous solution to be a combination of terms involving sin(t) and cos(t). Since your inhomogeneous term (-cos(t), sin(t)) is of the same functional form, the inhomogeneous solution can contain terms of the form

x= (A1, A2) cos(t) + (B1, B2) sin(t) + (C1, C2) t cos(t) + (D1, D2) t sin(t)

To determine the 8 unknown coefficients A1 through D2 you would substitute this x into your original equation and get 8 linear equations.

An easier method in this case is given by variation of parameters.

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## Comments

^{-1}. Then if b is the inhomogeneous term, the inhomogeneous part of the solution is given by^{-1}b dt.