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 coefficient matrix.