Systems of ODEs

An nxn system of ODEs looks like 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{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

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

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:

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 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.