Autonomous ODE - Revision history

Rice at 19:32, 17 May 2024

2024-05-17T19:32:50Z

← Older revision		Revision as of 19:32, 17 May 2024
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	Some equilibrium solutions are [[Equilibrium#Stability\|stable]], where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.		Some equilibrium solutions are [[Equilibrium#Stability\|stable]], where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.

			[[Category:Differential Equations]]

Rice at 23:39, 23 April 2024

2024-04-23T23:39:21Z

← Older revision		Revision as of 23:39, 23 April 2024
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	y' = F(y)		y' = F(y)
	</math>		</math>

			They can be solved by [[Separation of variables]] method.

	= Equilibrium Solutions =		= Equilibrium Solutions =

Rice: /* Equilibrium Analysis */

2024-04-15T23:14:12Z

Equilibrium Analysis

← Older revision		Revision as of 23:14, 15 April 2024
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	This can be visualized on a '''phase line'''.		This can be visualized on a '''phase line'''.

	Some equilibrium solutions are ''stable'', where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.		Some equilibrium solutions are [[Equilibrium#Stability\|stable]], where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.

Rice at 22:28, 15 April 2024

2024-04-15T22:28:36Z

← Older revision		Revision as of 22:28, 15 April 2024
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	</math>		</math>

	= Equilibrium =		= Equilibrium Solutions =

	Autonomous ODE's have trivial ODE solutions.		Autonomous ODE's have trivial ODE solutions.
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	</math>		</math>

	is ~~the~~ equilibrium solution of the ODE.		is an equilibrium solution of the ODE.

	If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math>		If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math>
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	</math>		</math>

	= Solution =		= General Solution =

	Autonomous equations can be solved by [[Separation of Variables]] method.		Autonomous equations can be solved by [[Separation of Variables]] method.

	= ~~Phase Line~~ =		= Equilibrium Analysis =

	Consider <math>y' = F(y)</math>		Consider <math>y' = F(y)</math>

	* If <math>F(y) = 0</math>, the solution is at equilibrium		* If <math>F(y) = 0</math>, the solution is at ''equilibrium''
	* If <math>F(y) >0</math>, then ''y'' is increasing in ''t''		* If <math>F(y) >0</math>, then ''y'' is increasing in ''t''
	* If <math>F(y) < 0</math>, then ''y'' is decreasing in ''t''		* If <math>F(y) < 0</math>, then ''y'' is decreasing in ''t''
			This can be visualized on a '''phase line'''.

			Some equilibrium solutions are ''stable'', where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.

Rice: Created page with "'''Autonomous ODE's''' have no explicit t-dependence. They come in the form $y' = F(y)$ = Equilibrium = Autonomous ODE's have trivial ODE solutions. If $F(c) = 0$ then $y(t) = c$ is the equilibrium solution of the ODE. If $y(t)$ is a solution, then so is $z(t) = y(t + t_0)$ for any constant $t_0$ $\begin{aligned} y'(t) &= F(y(t))\\ z'(t) &= y'(t + t_0) \\ &= F(y(t + t_0)) \\ &= F(z(t)..."$

2024-04-15T22:20:34Z

Created page with "'''Autonomous ODE's''' have no explicit t-dependence. They come in the form <math> y' = F(y) </math> = Equilibrium = Autonomous ODE's have trivial ODE solutions. If <math> F(c) = 0 </math> then <math> y(t) = c </math> is the equilibrium solution of the ODE. If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math> <math> \begin{aligned} y'(t) &= F(y(t))\\ z'(t) &= y'(t + t_0) \\ &= F(y(t + t_0)) \\ &= F(z(t)..."

New page

'''Autonomous ODE's''' have no explicit t-dependence. They come in the form

<math>
y' = F(y)
</math>

= Equilibrium =

Autonomous ODE's have trivial ODE solutions.

If

<math>
F(c) = 0
</math>

then

<math>
y(t) = c
</math>

is the equilibrium solution of the ODE.

If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math>

<math>
\begin{aligned}
y'(t) &= F(y(t))\\
z'(t) &= y'(t + t_0) \\
&= F(y(t + t_0)) \\
&= F(z(t))
\end{aligned}
</math>

= Solution =

Autonomous equations can be solved by [[Separation of Variables]] method.

= Phase Line =

Consider <math>y' = F(y)</math>

* If <math>F(y) = 0</math>, the solution is at equilibrium
* If <math>F(y) >0</math>, then ''y'' is increasing in ''t''
* If <math>F(y) < 0</math>, then ''y'' is decreasing in ''t''