solutions of ordinary differential equation

Let us consider the ordinary differential equation

$F(x,y,y^{\prime },y^{\prime \prime },\mathrm{\dots },y^{(n)})=0$(1)

of order $n$.

The general solution of (1) is a function

$$x\mapsto y=\phi (x,C_1,C_2,\mathrm{\dots },C_n)$$

satisfying the following conditions:

a) $y$ depends on $n$ arbitrary constants $C_1,C_2,\mathrm{\dots },C_n$.
b) $y$ satisfies (1) with all values of $C_1,C_2,\mathrm{\dots },C_n$
c) If there are given the initial conditions
    $y=y_0$,  $y^{\prime }=y_1$,  $y^{\prime \prime }=y_2$,  $\mathrm{\dots }$,  $y^{(n-1)}=y_{n-1}$ when $x=x_0,$
then one can chose the values of $C_1,C_2,\mathrm{\dots },C_n$ such that $y=\phi (x,C_1,C_2,\mathrm{\dots },C_n)$  fulfils those conditions (supposing that $x_0,y_0,y_1,y_2,\mathrm{\dots },y_{n-1}$ 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  $C_1,C_2,\mathrm{\dots },C_n$,  is called a particular solution of (1).