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

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:

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.

Initial value problem

If P(t) and g(t) are continuous, then the IVP

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{cases} \vec{x}'=P\vec{x}+g\\ \vec{x}=\vec{x}^{(0)} \end{cases} }

has a unique solution

If the results are linearly independent, it is a fundamental set of solutions for the ODE, i.e. they span the entire solution space.