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[parent] solutions of ordinary differential equation (Definition)

Let us consider the ordinary differential equation

$\displaystyle F(x,\,y,\,y',\,y'',\,\ldots,\,y^{(n)}) = 0$ (1)

of order $ n$.

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) $ y$ depends on $ n$ arbitrary constants $ C_1,\,C_2,\,\ldots,\,C_n$.
b) $ y$ satisfies (1) with all values of $ C_1,\,C_2,\,\ldots,\,C_n$
c) If there are given the initial conditions
                $ y = y_0$,$ y' = y_1$,$ y'' = y_2$, $ \ldots$, $ y^{(n-1)} = y_{n-1}$    when    $ x = x_0,$
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).



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"solutions of ordinary differential equation" is owned by pahio.
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Also defines:  general solution, particular solution

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general solution of linear differential equation (Result) by pahio
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Cross-references: solution, region, initial conditions, function, order, ordinary differential equation
There are 29 references to this entry.

This is version 1 of solutions of ordinary differential equation, born on 2007-01-05.
Object id is 8719, canonical name is SolutionsOfOrdinaryDifferentialEquation.
Accessed 3070 times total.

Classification:
AMS MSC: 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
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