$\begin{bmatrix} \dot x \\ \dot y \end{bmatrix}$$=$$\begin{bmatrix}-2 & 3 \\ -9 & 8 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix} $
has complex general solution
$\begin{bmatrix} x \\ y \end{bmatrix}$$=$$D_1\begin{bmatrix}5-\sqrt{2}i \\ 9 \end{bmatrix}e^{(3+\sqrt{2}i)}+D_2\begin{bmatrix}5+\sqrt{2}i \\ 9 \end{bmatrix}e^{(3-\sqrt{2}i)}$
Find the general real-valued solution.
Here is my attempt at it:
$e^{(3+\sqrt{2}i)}\begin{bmatrix}5-\sqrt{2}i \\ 9 \end{bmatrix}$= $e^{3t}[\cos(\sqrt{2}t)+i\sin(\sqrt{2}t)]\begin{bmatrix}5-\sqrt{2}i \\ 9 \end{bmatrix}$ =$e^{3t}\bigg(\begin{bmatrix}5\cos(\sqrt{2}t)+\sqrt2\sin(\sqrt{2}t) \\ 9\cos(\sqrt{2}t) \end{bmatrix}+i\begin{bmatrix}-\sqrt2\cos(\sqrt{2}t)+5\sin(\sqrt{2}t) \\ 9\sin(\sqrt{2}t) \end{bmatrix}\bigg)$
Hence, the real-valued solution $\begin{bmatrix} x \\ y \end{bmatrix}$=$ c_1e^{3t}\begin{bmatrix}5\cos(\sqrt{2}t)+\sqrt2\sin(\sqrt{2}t) \\ 9\cos(\sqrt{2}t) \end{bmatrix}+c_2e^{3t}\begin{bmatrix}-\sqrt2\cos(\sqrt{2}t)+5\sin(\sqrt{2}t) \\ 9\sin(\sqrt{2}t) \end{bmatrix}$
Can anyone verify this is the correct approach and solution.