An nxn system of ODEs looks like the following
In vector form, the same equation can be written as
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
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:
You can rewrite it as the following:
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
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
Its Wronskian would be correspondingly defined as the determinant.
Classifications