Ordinary differential equations Definition

Characteristic equation

Builds on Linear equation with constant coefficients

Definition

Let $a_0, \dots, a_{n-1}$ be real numbers, and consider the homogeneous linear equation with constant coefficients $y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = 0.$ Its characteristic polynomial is $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0$ . Its characteristic equation is the polynomial equation $p(r) = 0$ for an unknown complex number $r$ .

Remarks

For each complex $r$ , the function $y = e^{r x}$ satisfies $y^{(n)} + \dots + a_0 y = p(r)\, e^{r x}$ . So $e^{rx}$ is a solution if and only if $p(r) = 0$ . Example: the equation $y'' - 3 y' + 2 y = 0$ has the characteristic equation $r^2 - 3 r + 2 = 0$ , with roots $1$ and $2$ . The characteristic polynomial of a linear recurrence plays the same role for recurrences, with $r^k$ in place of $e^{r x}$ .

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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
Characteristic equation definition

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13 pages build on this one.

Tag	`0QDXMG46QW`
URI	`m2://0QDXMG46QW`
Aliases	`auxiliary-equation`
Kind	definition
Writing	W1 stated
MSC	34A30
Defines	Characteristic equation
Uses	Linear equation with constant coefficients

Characteristic equation

Definition

Remarks

Reading path

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