How resolved this difference equation used prediction methods? $$X' = \left[\begin{array}{ccc}1&-1&-2\\1&3&2\\1&-1&2\end{array}\right]X + \left[\begin{array}{c}t^{2}\\t+2\\2\end{array}\right]$$
Calculate eigenvalues and eigenvectors: $$\lambda_{1} = 2 + 2i$$ $$\lambda_{2} = 2-2i$$ $$\lambda_{3} = 2$$
$$v_{1} = (i,-i,1)$$ $$v_{2} = (-i,i,1)$$ $$v_{3} = (-1,-1,1)$$
So,
$$X_{b} = C_{1}\left(\begin{array}{c}i\\-i\\i\end{array}\right)e^{(2+2i)t} + C_{2} \left(\begin{array}{c}-i\\i\\1 \end{array}\right)e^{(2-2i)t} + C_{3} \left(\begin{array}{c}1\\-1\\ 1\end{array}\right)e^{2t}$$
How continue?
Hint: $$\begin {matrix} \dfrac {e^{i x} + e^{-i x}}{2} = ? \text { and } \dfrac {e^{i x} - e^{-i x}}{2} = ? \end {matrix}$$
(in terms of what functions?)
Also, $$\begin {matrix} \dfrac {e^{x} + e^{-x}}{2} = ? \text { and } \dfrac {e^{x} - e^{-x}}{2} = ? \end {matrix}$$
(in terms of what other functions?)