solutions of ordinary differential equation

Let us consider the ordinary differential equationMathworldPlanetmath

F(x,y,y,y′′,,y(n))=0 (1)

of order n.

The general solution of (1) is a function


satisfying the following conditions:

a) y depends on n arbitrary constants C1,C2,,Cn.
b) y satisfies (1) with all values of C1,C2,,Cn
c) If there are given the initial conditionsMathworldPlanetmath
    y=y0,  y=y1,  y′′=y2,   ,  y(n-1)=yn-1 when x=x0,
then one can chose the values of C1,C2,,Cn such that  y=φ(x,C1,C2,,Cn)  fulfils those conditions (supposing that x0,y0,y1,y2,,yn-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  C1,C2,,Cn,  is called a particular solution of (1).

Title solutions of ordinary differential equation
Canonical name SolutionsOfOrdinaryDifferentialEquation
Date of creation 2013-03-22 16:32:16
Last modified on 2013-03-22 16:32:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Definition
Classification msc 34A05
Related topic DerivativesOfSolutionOfFirstOrderODE
Defines general solution
Defines particular solution