Systems of ODEs: Difference between revisions
(2 intermediate revisions by the same user not shown) | |||
Line 72: | Line 72: | ||
has a unique solution | has a unique solution | ||
If the results are [[linear independence (functions)|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 | |||
<math> | |||
\vec{x}(t)=c_1\vec{x}^{(1)}+\ldots+c_k\vec{x}^{(k)}=X\vec{c} | |||
</math> | |||
Its [[Wronskian]] would be correspondingly defined as the determinant. | |||
= Classifications = | |||
* [[Homogeneous linear systems: constant coefficients]] |
Latest revision as of 15:01, 10 June 2024
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.