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# Matrix form: Inhomogeneous differential equations

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How do I express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form?
(see the attachment for the full question)
x = -2x - y + 12t + 12,
y = 2x - 5y - 5

How do I express the corresponding homogeneous system of differential equations, also in matrix form?

How do I find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. From this how would I write down the complementary function for the system of differential equations?

How would you calculate a particular integral for the inhomogeneous system, and then find the general solution?

How would I determine the particular solution of the initial-value problem with the initial conditions x(0) = 3 and y(0) = 2?

I have another problem below but on a similar topic:

If an object moves in the plane in such a way that its Cartesian coordinates (x, y) at time t satisfy the following homogeneous system of second-order differential equations:
x = -2x - y,
y = 2x - 5y.

How would I:
Express the system in matrix form?
Find the general solution of the system?
I think this system undergoes simple harmonic motion in a straight line in two distinct ways but why?
And for each such simple harmonic motion how do I determine the angular frequency and the vector giving the direction of motion?