# homogeneous linear differential equation

The linear differential equation

${a}_{n}(x){y}^{(n)}+{a}_{n-1}(x){y}^{(n-1)}+\mathrm{\dots}+{a}_{1}(x){y}^{\prime}+{a}_{0}(x)y=b(x)$ | (1) |

is called
homogeneous^{} (http://planetmath.org/HomogeneousLinearDifferentialEquation)
iff $b(x)\equiv 0$. If $b(x)\not\equiv 0$,
the equation (1) is inhomogeneous.

If (1) is homogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation),
then the sum of any solutions is a solution and any solution
multiplied by a constant is a solution.

The special case

$${c}_{n}{x}^{n}{y}^{(n)}+{c}_{n-1}{x}^{n-1}{y}^{(n-1)}+\mathrm{\dots}+{c}_{1}x{y}^{\prime}+{c}_{0}y=\mathrm{\hspace{0.33em}0}$$ |

of (1), where the ${c}_{i}$’s are constants, can via the
substitution $x={e}^{t}$ be transformed into a homogeneous
linear differential equation of the same order but with
constant coefficients.

Title | homogeneous linear differential equation |
---|---|

Canonical name | HomogeneousLinearDifferentialEquation |

Date of creation | 2014-02-27 10:07:04 |

Last modified on | 2014-02-27 10:07:04 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 3 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 34A05 |