# 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).

Title solutions of ordinary differential equation SolutionsOfOrdinaryDifferentialEquation 2013-03-22 16:32:16 2013-03-22 16:32:16 pahio (2872) pahio (2872) 5 pahio (2872) Definition msc 34A05 DerivativesOfSolutionOfFirstOrderODE general solution particular solution