First-order first-degree autonomous differential equation - Revision history

Vipul: /* Analysis */

2012-07-05T21:51:22Z

Analysis

← Older revision		Revision as of 21:51, 5 July 2012
Line 53:		Line 53:
	\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>		\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>
	\|-		\|-
	\| linear times logarithmic, something like <math>x (\ln x + 1)</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math> (note: if we wanted growth between exponential and double exponential, we would need something like <math>x(\~~lnx~~ + 1)^\gamma, 0 < \gamma < 1</math>, and if we wanted triple exponential growth, we would multiply by a double logarithmic term)		\| linear times logarithmic, something like <math>x (\ln x + 1)</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math> (note: if we wanted growth between exponential and double exponential, we would need something like <math>x(\ln x + 1)^\gamma, 0 < \gamma < 1</math>, and if we wanted triple exponential growth, we would multiply by a double logarithmic term)
	\|-		\|-
	\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.		\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.
	\|}		\|}

Vipul: /* Starting at time zero with value one */

2012-07-05T21:51:10Z

Starting at time zero with value one

← Older revision		Revision as of 21:51, 5 July 2012
Line 51:		Line 51:
	\| <math>f(x) := x^\gamma, 0 < \gamma < 1</math> \|\| <math>x = O(t^{1/(1 - \gamma)})</math> (i.e., it grows roughly like a power function of <math>t</math>)		\| <math>f(x) := x^\gamma, 0 < \gamma < 1</math> \|\| <math>x = O(t^{1/(1 - \gamma)})</math> (i.e., it grows roughly like a power function of <math>t</math>)
	\|-		\|-
	\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>		\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>
	\|-		\|-
	\| linear times logarithmic, something like <math>x (\ln x + 1)</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math>		\| linear times logarithmic, something like <math>x (\ln x + 1)</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math> (note: if we wanted growth between exponential and double exponential, we would need something like <math>x(\lnx + 1)^\gamma, 0 < \gamma < 1</math>, and if we wanted triple exponential growth, we would multiply by a double logarithmic term)
	\|-		\|-
	\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.		\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.
	\|}		\|}

Vipul: /* Analysis */

2012-07-05T21:49:52Z

Analysis

← Older revision		Revision as of 21:49, 5 July 2012
Line 42:		Line 42:
	subject to the initial condition that at <math>t = 0</math>, <math>x = 1</math>.		subject to the initial condition that at <math>t = 0</math>, <math>x = 1</math>.

	We make cases based on the growth rate of <math>f</math>:		We consider various possibilities for <math>f</math> a function that sends 1 and higher numbers to positive numbers, and make cases based on the growth rate of <math>f</math>:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Starting at time zero with value one */

2012-07-05T15:19:54Z

Starting at time zero with value one

← Older revision		Revision as of 15:19, 5 July 2012
Line 49:		Line 49:
	\| constant function <math>f(x) := c</math> \|\| linear function <math>x := 1 + ct</math>		\| constant function <math>f(x) := c</math> \|\| linear function <math>x := 1 + ct</math>
	\|-		\|-
	\| <math>f(x) := x^\gamma, 0 < \gamma < 1</math> \|\| <math>x = O(t^{1/(1 - \gamma)}</math> (i.e., it grows roughly like a power function of <math>t</math>)		\| <math>f(x) := x^\gamma, 0 < \gamma < 1</math> \|\| <math>x = O(t^{1/(1 - \gamma)})</math> (i.e., it grows roughly like a power function of <math>t</math>)
	\|-		\|-
	\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>		\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>

Vipul: /* Starting at time zero with value one */

2012-07-04T16:50:50Z

Starting at time zero with value one

← Older revision		Revision as of 16:50, 4 July 2012
Line 53:		Line 53:
	\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>		\| linear function <math>f(x) := mx, m > 0</math> \|\| exponential function <math>x := \exp(mt)</math>
	\|-		\|-
	\| linear times logarithmic, something like <math>x \ln x</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math>		\| linear times logarithmic, something like <math>x (\ln x + 1)</math> \|\| <math>x</math> grows something like a doubly exponential function of <math>t</math>
	\|-		\|-
	\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.		\| <math>f(x) := x^\gamma, \gamma > 1</math> \|\| <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.
	\|}		\|}

Vipul: Vipul moved page First-order first-degree autonomous differential equation of degree one to First-order first-degree autonomous differential equation

2012-07-01T19:10:40Z

Vipul moved page First-order first-degree autonomous differential equation of degree one to First-order first-degree autonomous differential equation

← Older revision	Revision as of 19:10, 1 July 2012
(No difference)

Vipul: Vipul moved page First-order autonomous differential equation of degree one to First-order first-degree autonomous differential equation of degree one

2012-07-01T19:09:48Z

Vipul moved page First-order autonomous differential equation of degree one to First-order first-degree autonomous differential equation of degree one

← Older revision	Revision as of 19:09, 1 July 2012
(No difference)

Vipul: Created page with "==Definition== Following the convention for autonomous differential equation, we denote the dependent variable by $x$ and independent variable by $t$
2012-06-30T22:02:55Z

Created page with "==Definition== Following the convention for autonomous differential equation, we denote the dependent variable by <math>x</math> and independent variable by <math>t</math..."

New page
==Definition==

Following the convention for [[autonomous differential equation]], we denote the dependent variable by <math>x</math> and independent variable by <math>t</math>.

===Form of the differential equation===

The differential equation is of the form:

<math>\frac{dx}{dt} = f(x)</math>

===Solution method and formula===

We convert the differential equation to an integration problem:

<math>\int \frac{dx}{f(x)} = \int dt</math>

and carry out the integrations on both sides. If <math>p</math> is an antiderivative of <math>1/f</math>, the solution will be:

<math>p(x) = t + C</math>

with <math>C</math> the freely varying parameter over <math>\R</math>: every particular value of <math>C</math> gives a solution. To express <math>x</math> as a function of <math>t</math>, we need to invert <math>p</math>. If we can do so, we'd have:

<math>x = p^{-1}(t + C)</math>

as the general solution.

In addition, there may be stationary solutions. These are solutions that correspond to constant functions <math>x = k</math> that satisfy <math>f(k) = 0</math>.

==Related notions==

* [[Separable differential equation]]: A slightly more general type of [[first-order differential equation]].
* [[Second-order autonomous differential equation of degree one]]: Although such equations cannot always be solved, they ''can'' always be reduced to [[first-order differential equation]]s.

==Analysis==

===Starting at time zero with value one===

Suppose we want to solve the initial value problem for the differential equation:

<math>\frac{dx}{dt} = f(x)</math>

subject to the initial condition that at <math>t = 0</math>, <math>x = 1</math>.

We make cases based on the growth rate of <math>f</math>:

{| class="sortable" border="1"
! Nature of <math>f</math> !! Nature of <math>x</math> in terms of <math>t</math>?
|-
| constant function <math>f(x) := c</math> || linear function <math>x := 1 + ct</math>
|-
| <math>f(x) := x^\gamma, 0 < \gamma < 1</math> || <math>x = O(t^{1/(1 - \gamma)}</math> (i.e., it grows roughly like a power function of <math>t</math>)
|-
| linear function <math>f(x) := mx, m > 0</math> || exponential function <math>x := \exp(mt)</math>
|-
| linear times logarithmic, something like <math>x \ln x</math> || <math>x</math> grows something like a doubly exponential function of <math>t</math>
|-
| <math>f(x) := x^\gamma, \gamma > 1</math> || <math>x</math> grows so fast in terms of <math>t</math> that it reaches <math>\infty</math> in finite time.
|}