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Homesolutions of ordinary differential equation

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# solutions of ordinary differential equation

Let us consider the ordinary differential equation

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

of order $n$.

The 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

$y=y_{0}$, $y^{{\prime}}=y_{1}$, $y^{{\prime\prime}}=y_{2}$,
$\ldots$, $y^{{(n-1)}}=y_{{n-1}}$ 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).

## Mathematics Subject Classification

34A05*no label found*

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