Back to the index. Or to the chambers

This article has 6 links. View as Cloud or List.

Partial Order Equation Function Polygon Differential Equation Initial Value Problem

1	Partial Order
1	Polygon
1	Function
1	Equation
2	Differential Equation
4	Initial Value Problem

Loading ...

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Solutions Of Ordinary Differential Equation

Let us consider the ordinary differential equation

$\displaystyle F(x,\,y,\,y',\,y'',\,\ldots,\,y^{(n)}) = 0$

(1)

of order

The general solution of (1) is a function

$\displaystyle x\mapsto y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$

satisfying the following conditions:
a)

depends on

arbitrary constants $C_1,\,C_2,\,\ldots,\,C_n$ .
b)

satisfies (1) with all values of $C_1,\,C_2,\,\ldots,\,C_n$
c) If there are given the initial conditions

, $\ldots$ , $y^{(n-1)} = y_{n-1}$ when

then one can chose the values of $C_1,\,C_2,\,\ldots,\,C_n$ such that $y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$ fulfils those conditions (supposing that $x_0,\,y_0,\,y_1,\,y_2,\,\ldots,\,y_{n-1}$ belong to the region where the conditions for the existence of the solution are valid).

Each function which is obtained from the general solution by giving certain concrete values for $C_1,\,C_2,\,\ldots,\,C_n$ , is called a particular solution of (1).