Homogeneous linear systems: constant coefficients - Revision history

Rice at 17:15, 10 June 2024

2024-06-10T17:15:29Z

← Older revision		Revision as of 17:15, 10 June 2024
Line 19:		Line 19:
	</math>		</math>

	Notice that \xi is an [[eigenvector]]. The entire eigenspace is solutions.		Notice that \xi is an [[eigenvector]]. The entire eigenspace is solutions. As such, we use the characteristic polynomial

			<math>
			A-\lambda I
			</math>

			where lambda is the eigenvalues.

			There are cases where there are repeated roots.

			There will be a general solution if A is [[diagonalization\|diagonalizable]] and n eigenvectors are linearly independent and form a basis in R^n, where we get the fundamental set

			<math>
			e^{r_1t}\xi^{(1)}, \ldots, e^{r_nt}\xi^{(n)}
			</math>

			The [[Wronskian]] is then

			<math>
			e^{t\sum r}\|\xi^{(1)}, \xi^{(1)}\|
			</math>

Rice at 15:11, 10 June 2024

2024-06-10T15:11:20Z

← Older revision		Revision as of 15:11, 10 June 2024
Line 18:		Line 18:
	r\vec{\xi}=A\vec{\xi}		r\vec{\xi}=A\vec{\xi}
	</math>		</math>

			Notice that \xi is an [[eigenvector]]. The entire eigenspace is solutions.

Rice at 15:09, 10 June 2024

2024-06-10T15:09:18Z

← Older revision		Revision as of 15:09, 10 June 2024
Line 11:		Line 11:
	<math>		<math>
	\vec{x}=e^{rt}\vec{\xi }		\vec{x}=e^{rt}\vec{\xi }
			</math>

			Doing some basic differentiation and substitution from the above two equations, we get

			<math>
			r\vec{\xi}=A\vec{\xi}
	</math>		</math>

Rice: Created page with "Category:Differential Equations Homogeneous linear systems of ODEs take the following form: $\vec{x}'=Ax$ Similar to the scalar case, we look for some exponential solutions: $\vec{x}=e^{rt}\vec{\xi }$ "

2024-06-10T15:06:55Z

Created page with "Category:Differential Equations Homogeneous linear systems of ODEs take the following form: <math> \vec{x}'=Ax </math> Similar to the scalar case, we look for some exponential solutions: <math> \vec{x}=e^{rt}\vec{\xi } </math>"

New page

[[Category:Differential Equations]]

Homogeneous linear systems of [[ODE]]s take the following form:

<math>
\vec{x}'=Ax
</math>

Similar to the scalar case, we look for some exponential solutions:

<math>
\vec{x}=e^{rt}\vec{\xi }
</math>

Homogeneous linear systems: constant coefficients - Revision history

Rice at 17:15, 10 June 2024

Rice at 15:11, 10 June 2024

Rice at 15:09, 10 June 2024

Rice: Created page with "Category:Differential Equations Homogeneous linear systems of ODEs take the following form: \vec{x}'=Ax Similar to the scalar case, we look for some exponential solutions: \vec{x}=e^{rt}\vec{\xi } "

Rice: Created page with "Category:Differential Equations Homogeneous linear systems of ODEs take the following form: $\vec{x}'=Ax$ Similar to the scalar case, we look for some exponential solutions: $\vec{x}=e^{rt}\vec{\xi }$ "