Linear System of Differential Equations, Solutions, Phase Portrait Sketching

Solutions of Homogeneous Linear System of DE
$\displaystyle \mathbf{y}'=\mathbf{A}\mathbf{y}$
$\displaystyle \mathbf{y}(t)=\mathbf{v}e^{rt}$
where $r$ and $\mathbf{v}$ are eigenvalue and eigenvector for $\mathbf{A}$ respectively.

Superposition Principle
If $\mathbf{x_1}(t)$ and $\mathbf{x_2}(t)$ are two solutions to a homogeneous SDE $\mathbf{y'}=\mathbf{Ay}$ , then $\displaystyle \mathbf{y}=c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)$ is also a solution for any scalars $c_1$ , $c_2$ .

Euler’s formula
$\displaystyle e^{i\theta}=\cos\theta+i\sin\theta$

General Solutions (Complex Eigenvalues)

1) Let $r_1=a+bi$ be an eigenvalue corresponding to eigenvector $\mathbf{v_1}$ . (The eigenvectors are complex conjugates: $\mathbf{v_1,v_2}=\mathbf{p}\pm \mathbf{q} i$ .)
2) Construct
$\displaystyle \mathbf{x}_\text{Re}(t)=e^{at}(\mathbf{p}\cos bt-\mathbf{q}\sin bt)$
$\displaystyle \mathbf{x}_\text{Im}(t)=e^{at}(\mathbf{p}\sin bt+\mathbf{q}\cos bt)$
3) The general solution is $\displaystyle \mathbf{y}=c_1\mathbf{x}_\text{Re}(t)+c_2\mathbf{x}_\text{Im}(t).$