Linear Ordinary Differential Equations |

where are all functions of . This differential equation is |

Linear Dependence and Independence |

In other words, if one of these functions can be expressed in terms (by linear combination) of others, these functions are linearly dependent on the interval .
In other words, if none of these functions can be expressed in terms (by linear combination) of others, these functions are linearly independent on the interval .
where is the determinant. If the Wronskian is |

Solutions and Superposition |

If are solutions of this where .
If are n where are arbitrary constants.
where . If contains no arbitrary constants and satisfies this differential equation, i.e., is called the
where . If is the particular solution and , the complementary solution, is the general solution of the associated homogeneous differential equation then the where are arbitrary constants, are n independent solutions of the associated homogeneous equation. |