Consider the Second-order System $$ \dot{\mathbf{x}} = A \mathbf{x} $$
$a_1$ and $a_2$ are positive real numbers, with $a_1 > 2*a_2$.
The matrix A is given as $$A = \begin{pmatrix} 0 & 1 \\ -a_2 & -a_1 \\ \end{pmatrix} $$
The values $a_1$ and $a_2$ are chosen so that the Matrix $A$ has negative real eigenvectors.
What I am really interested in: Is there an analytic (parametric) formula for the state-space trajectories of this system?
Yes. Verify by differentiation $$x(t) = e^{A(t-t_0)}x(t_0).$$ This formula is true regardless of the eigenvalues of the system but is obviously unbounded if the eigenvalues are unstable. To get the matrix exponential you can solve for the eigenvalues parametrically and compute everything analytically for this 2x2 case.