Eigenvalue Method for Homogeneous Systems

For a homogeneous system, you have $D_x \vec{x}=A \vec{x}$. If you can find the eigenvalues $\lambda_1, \cdots, \lambda_n$ and eigenvectors $\vec{v_1}, \cdots, \vec{v_n}$, then $\vec{x}= \sum\limits_{i=1}^n c_i e^{\lambda_i} \vec{v_i}$

If there are repeated eigenvalues, solve for the known one like normal: (In this example I'm just showing for a system of two, but it extends) $\vec{x}=\vec{x_1} + \vec{x_2}$, $\vec{x_1}=c_1 \vec{v_1} e^{\lambda_1 t}$, let $\vec{x_2}=\vec{w} t e^t \Rightarrow D_t \vec{x_2} = \vec{w} (e^t + te^t)$. Then plug back into initial D.E., and solve for $\vec{w}$.