Systems of ODEs: Difference between revisions
(Created page with "Category:Differential Equations An nxn system of ODEs looks like the following <math> \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} </math>") |
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An nxn system of [[ODE]]s looks like the following | An nxn system of [[ODE]]s looks like the following | ||
<math> | <math> | ||
\begin{aligned} | \begin{aligned} | ||
| Line 10: | Line 11: | ||
\end{aligned} | \end{aligned} | ||
</math> | </math> | ||
In vector form, the same equation can be written as | |||
<math> | |||
\vec{x}'=\vec{F}(t,\vec{x}) | |||
</math> | |||
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 | |||
<math> | |||
\vec{x}'=P(t)\vec{x}+g(t) | |||
</math> | |||
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: | |||
<math> | |||
y'' + py' + qy = g | |||
</math> | |||
You can rewrite it as the following: | |||
<math> | |||
\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} | |||
</math> | |||
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 | |||
<math> | |||
\begin{cases} | |||
\vec{x}'=P\vec{x}+g\\ | |||
\vec{x}=\vec{x}^{(0)} | |||
\end{cases} | |||
</math> | |||
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
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, where all solutions can be described 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}(t)=c_1\vec{x}^{(1)}+\ldots+c_k\vec{x}^{(k)}=X\vec{c} }
Its Wronskian would be correspondingly defined as the determinant.
