The state equation of a second order system is
$\dot{\boldsymbol{x}}(t)=\mathrm{A} \boldsymbol{x}(t), \; \boldsymbol{x}(0)$ is the initial condition.
Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $\mathrm{A}$ and $v_1$ and $v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution, $\boldsymbol{x}(t)$, of the state equation is
- $\sum_{i=1}^2 \alpha_i e^{\lambda_i \mathrm{t}} \boldsymbol{v}_i$
- $\sum_{i=1}^2 \alpha_i e^{2 \lambda_i \mathrm{t}} \boldsymbol{v}_i$
- $\sum_{i=1}^2 \alpha_i e^{3 \lambda_i \mathrm{t}} \boldsymbol{v}_i$
- $\sum_{i=1}^2 \alpha_i e^{4 \lambda_i \mathrm{t}} \boldsymbol{v}_i$