Particular solution to nonhomogeneous system with imaginary eigenvalues

[Klasj]

I have the system:

I have found the fundamental matrix to be:

Based on the eigenvalues λ = +- 4i
Which gives me the homogeneous solution:

Now I need to find the particular solution because I have the matrix [2 -1] added to my system. I have tried to google how to do this but since my eigenvalues are imaginary I can't seem to find help.
The excercise also asks me to calculate the inverse of the fundamental matrix, then multiply it with [2 -1] and integrate but I am not sure how this can help me find the particular solution.
Any help or advice would be appreciated!

 

[HallsofIvy]

I don't see that the eigenvalues have anything to do with finding the particular solutions. Since the right hand side of the equation is a constant matrix, the specific solution will be a constant matrix. Try \(\displaystyle y= \begin{bmatrix}A \\ B\end{bmatrix}\) where A and B are constants. Then \(\displaystyle y'= \begin{bmatrix}0 \\ 0 \end{bmatrix}\). The equation becomes \(\displaystyle \begin{bmatrix}0 \\ 0\end{bmatrix}= \begin{bmatrix}-4 & 4 \\-8 & 4 \end{bmatrix}\begin{bmatrix}A \\ B \end{bmatrix}+ \begin{bmatrix} 2 \\ -1 \end{bmatrix}= \begin{bmatrix}-4A+ 4B+ 2 \\ -8A+ 4B- 1\end{bmatrix}\).

Solve -4A+ 4B+ 2= 0 and -8A+ 4B- 1= 0.