Ordinary differential equations Definition

Linear equation with constant coefficients

Builds on Linear differential equation of order n

Definition

Let $n \ge 1$ , let $a_0, \dots, a_{n-1}$ be real numbers, let $I$ be an open interval and let $b : I \to \mathbb{R}$ be continuous. A linear equation with constant coefficients is a linear differential equation of the form $y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = b(x).$ Its homogeneous equation is the case $b = 0$ , on the interval $I = \mathbb{R}$ .

Remarks

The function $e^{r x}$ solves the homogeneous equation if and only if $r$ is a root of the characteristic equation $r^n + a_{n-1} r^{n-1} + \dots + a_0 = 0$ . The roots of this polynomial give a fundamental system. See the cases of distinct real roots, repeated roots and complex conjugate roots. The harmonic oscillator $x'' + \omega^2 x = 0$ is an example of order $2$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
Linear equation with constant coefficients definition

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Tag	`0A6PGTTP2A`
URI	`m2://0A6PGTTP2A`
Kind	definition
Writing	W0 stub
MSC	34A30
Defines	Linear equation with constant coefficients
Uses	Linear differential equation of order n

Linear equation with constant coefficients

Definition

Remarks

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