# 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).

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 |