Let us consider the ordinary differential equation

(1)

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
                $y = y_0$ ,$y' = y_1$ ,$y'' = y_2$ , $\ldots$ ,$y^{(n-1)} = y_{n-1}$     when-13-JG
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).