Ordinary differential equation - Revision history

Rice: /* Solvable Classes */

2024-06-10T06:51:43Z

Solvable Classes

2024-05-01T21:18:24Z

Solvable Classes

← Older revision		Revision as of 21:18, 1 May 2024
Line 53:		Line 53:
	*[[First Order Scalar ODE\|First Order Scalar]]		*[[First Order Scalar ODE\|First Order Scalar]]
	*[[Autonomous ODE]]		*[[Autonomous ODE]]
			* [[Second order linear ODE]]

	= Applications =		= Applications =

2024-04-17T21:38:43Z

Solving ODE

← Older revision		Revision as of 21:38, 17 April 2024
Line 26:		Line 26:

	= Solving ODE =		= Solving ODE =
	An ODE is considered ''solve'' if we can express ''y'' in terms of integrals of explicitly known functions. We don't necessarily need to solve the integral; some integrals are very very hard.		An ODE is considered ''solve'' if we can express ''y'' in terms of integrals of explicitly known functions (reduce to quadratures). We don't necessarily need to solve the integral; some integrals are very very hard; MATLAB just does it.

	= Classification =		= Classification =

2024-04-17T21:37:57Z

Classification

← Older revision		Revision as of 21:37, 17 April 2024
Line 7:		Line 7:
	''Ordinary'' means that the equation has one variable, as opposed to		''Ordinary'' means that the equation has one variable, as opposed to
	partial differential.		partial differential.

	~~There is ''no general method'' to solve ODEs. We separate them by classes~~
	~~and solve them individually.~~

	== Example ==		== Example ==
Line 32:		Line 29:

	= Classification =		= Classification =
			There is ''no general method'' to solve ODEs. We separate them by classes
			and solve them individually.

	== Linearity ==		== Linearity ==

2024-04-17T21:35:22Z

Example

← Older revision		Revision as of 21:35, 17 April 2024
Line 27:		Line 27:
	Notably, the solution is ''homogeneous'', meaning that <math>y(t) = 0</math> is		Notably, the solution is ''homogeneous'', meaning that <math>y(t) = 0</math> is
	a solution. This will probably be covered later.		a solution. This will probably be covered later.

			= Solving ODE =
			An ODE is considered ''solve'' if we can express ''y'' in terms of integrals of explicitly known functions. We don't necessarily need to solve the integral; some integrals are very very hard.

	= Classification =		= Classification =

2024-04-17T21:33:00Z

← Older revision	Revision as of 21:33, 17 April 2024
(No difference)

2024-04-15T23:14:16Z

Solvable Classes

← Older revision		Revision as of 23:14, 15 April 2024
Line 50:		Line 50:
	* [[Linear First Order ODE]]		* [[Linear First Order ODE]]
	*[[First Order Scalar ODE\|First Order Scalar]]		*[[First Order Scalar ODE\|First Order Scalar]]
			*[[Autonomous ODE]]

2024-04-15T21:44:20Z

Solvable Classes

← Older revision		Revision as of 21:44, 15 April 2024
Line 49:		Line 49:
	Here is a list of ODEs we study and solve, from simple to complex:		Here is a list of ODEs we study and solve, from simple to complex:
	* [[Linear First Order ODE]]		* [[Linear First Order ODE]]
	*		*[[First Order Scalar ODE\|First Order Scalar]]

2024-04-15T21:35:15Z

Other Classifications

← Older revision		Revision as of 21:35, 15 April 2024
Line 38:		Line 38:
	construct solutions. As one see from [[Linear Algebra]], for a linear problem, a general solution can be constructed from ''one specific solution'' and the ''homogeneous solution''.		construct solutions. As one see from [[Linear Algebra]], for a linear problem, a general solution can be constructed from ''one specific solution'' and the ''homogeneous solution''.

	== ~~Other Classifications~~ ==		== Order ==
	The '''order''' of an ODE is the order of its highest derivative.		The '''order''' of an ODE is the order of its highest derivative. Higher order ODEs are generally harder to solve.

			== Scalar/System ==
	In a '''scalar''', there is only one unknown function <math>y(t)</math>.		In a '''scalar''', there is only one unknown function <math>y(t)</math>.
	In a '''system''', there are several, and you have to solve them		In a '''system''', there are several, and you have to solve them
	simultaneously.		simultaneously (such as predator-prey functions).

	== Solvable Classes ==		== Solvable Classes ==

2024-04-15T21:31:26Z

Classification

← Older revision		Revision as of 21:31, 15 April 2024
Line 30:		Line 30:
	= Classification =		= Classification =

			== Linearity ==
	An ODE is '''linear''' if all terms are proportional to <math>y, y',		An ODE is '''linear''' if all terms are proportional to <math>y, y',
	y''. \ldots</math> or are given functions of <math>t</math>. This		y''. \ldots</math> or are given functions of <math>t</math>.

			This
	distinction is especially useful since linear combination can be used to		distinction is especially useful since linear combination can be used to
	construct solutions.		construct solutions. As one see from [[Linear Algebra]], for a linear problem, a general solution can be constructed from ''one specific solution'' and the ''homogeneous solution''.

			== Other Classifications ==
	The '''order''' of an ODE is the order of its highest derivative.		The '''order''' of an ODE is the order of its highest derivative.

Line 41:		Line 45:
	simultaneously.		simultaneously.

	Here is a list of ODEs we study, from simple to complex:		== Solvable Classes ==
			Here is a list of ODEs we study and solve, from simple to complex:
	* [[Linear First Order ODE]]		* [[Linear First Order ODE]]
			*


	= Applications =		= Applications =