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

${\vec {x}}'={\vec {F}}(t,{\vec {x}})$

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

${\vec {x}}'=P(t){\vec {x}}+g(t)$

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} }

First order systems are more general than higher order scalars, in that not all first order systems can be represented as a scalar.