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:

${\begin{pmatrix}y_{1}'\\y_{2}'\end{pmatrix}}={\begin{pmatrix}0&1\\-q&-p\end{pmatrix}}{\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}+{\begin{pmatrix}0\\g\end{pmatrix}}$

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

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

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, where all solutions can be described as

${\vec {x}}(t)=c_{1}{\vec {x}}^{(1)}+\ldots +c_{k}{\vec {x}}^{(k)}=X{\vec {c}}$

Its Wronskian would be correspondingly defined as the determinant.