solutions of ordinary differential equationSolutionsOfOrdinaryDifferentialEquation2007-01-05 17:51:302009-08-04 10:23:57Definitiondifferential equationgeneral solutionparticular solution% this is the default PlanetMath preamble. as your knowledge
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Let us consider the ordinary differential equation
\begin{align}
F(x,\,y,\,y',\,y'',\,\ldots,\,y^{(n)}) = 0
\end{align}
of order $n$.
The {\em general solution} of (1) is a function
$$x\mapsto y = \varphi(x,\,C_1,\,C_2,\,\ldots,\,C_n)$$
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\\
\qquad\qquad $y = y_0$,\,\,$y' = y_1$,\,\,$y'' = y_2$,\,\,
$\ldots$,\,\,$y^{(n-1)} = y_{n-1}$\quad when\quad$x = x_0,$\\
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 {\em particular solution} of (1).