Systems of ODEs - Revision history

Rice: /* Initial value problem */

2024-06-10T15:01:36Z

Initial value problem

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	Its [[Wronskian]] would be correspondingly defined as the determinant.		Its [[Wronskian]] would be correspondingly defined as the determinant.

			= Classifications =
			* [[Homogeneous linear systems: constant coefficients]]

2024-06-10T14:36:29Z

Initial value problem

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	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.		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.

2024-06-10T14:30:18Z

Initial value problem

← Older revision		Revision as of 14:30, 10 June 2024
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	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.

2024-06-10T14:15:54Z

Initial value problem

2024-06-10T14:14:41Z

Reduce order

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	First order systems are more general than higher order scalars, in that not all first order systems can be represented as a scalar.		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 solution

2024-06-10T14:09:23Z

Reduce order

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	\end{pmatrix}		\end{pmatrix}
	</math>		</math>

			First order systems are more general than higher order scalars, in that not all first order systems can be represented as a scalar.

2024-06-10T14:07:58Z

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	where P is the nxn coefficient matrix.		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>

2024-06-10T07:00:03Z

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	</math>		</math>

	where P is the coefficient matrix.		where P is the nxn coefficient matrix.

2024-06-10T06:59:31Z

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	</math>		</math>

	Non-linear ODEs are very complicated (like chaos theory, three body problem, etc.)		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 coefficient matrix.

2024-06-10T06:57:04Z

← Older revision		Revision as of 06:57, 10 June 2024
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	\vec{x}'=\vec{F}(t,\vec{x})		\vec{x}'=\vec{F}(t,\vec{x})
	</math>		</math>

			Non-linear ODEs are very complicated (like chaos theory, three body problem, etc.)