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).

How to Sketch Phase Portrait

Probably the best video on how to sketch Phase Portrait:

Author: mathtuition88

Math and Education Blog

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