Systems of ODEs

An nxn system of ODEs looks like the following

${\begin{aligned}x_{1}'&=F_{1}(t,x_{1},x_{2},\ldots ,x_{n})\\x_{2}'&=F_{2}(t,x_{1},x_{2},\ldots ,x_{n})\\\ldots \\x_{n}'&=F_{n}(t,x_{1},x_{2},\ldots ,x_{n})\\\end{aligned}}$

In vector form, the same equation can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}'=\vec{F}(t,\vec{x}) }

Non-linear ODEs are very complicated (like chaos theory, three body problem, etc.). This page will focus entirely on linear systems, which can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}'=P(t)\vec{x}+g(t) }

where P is the nxn coefficient matrix.

Reduce order

Given a higher order ODE, one can write it as a system of first order ODEs.

For example, take the following second order linear ODE:

$y''+py'+qy=g$

You can rewrite it as the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} y_1' \\ y_2' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -q & -p \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} + \begin{pmatrix} 0 \\ g \end{pmatrix} }