Given the following system of differential equations:\begin{align*} \begin{cases} x_1'=3x_1+x_2 \\ x_2'=2x_1+x_2 \end{cases} \end{align*}
Putting into standard form: $$\vec x'= \begin{bmatrix} 3&1\\ 2&1 \end{bmatrix}\vec x $$The eigenvalues are $\lambda=2\pm{\sqrt3}$
And thus the associated eigenvectors are:
$$ \begin{bmatrix} -\frac{{-1-\sqrt3}}{2}\\ 1 \end{bmatrix} , \begin{bmatrix} -\frac{{\sqrt3}-1}{2}\\ 1 \end{bmatrix} $$ Would this, then, be the correct general form? $$\vec x=c_1\begin{bmatrix} -\frac{{-1-\sqrt3}}{2}\\ 1 \end{bmatrix}e^{(2+{\sqrt3})t}+c_2\begin{bmatrix} -\frac{{\sqrt3}-1}{2}\\ 1 \end{bmatrix}e^{(2-{\sqrt3})t}$$