Ordinary differential equations Linear equations and systems Theorem

Abel's identity

Builds on Wronskian Homogeneous linear equation

Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, and let $y_1, y_2$ be solutions on $I$ of $y'' + p(x)\, y' + q(x)\, y = 0.$ Let $W = y_1 y_2' - y_1' y_2$ be their Wronskian, and let $x_0 \in I$ . Then for each $x \in I$ , $W(x) = W(x_0) \exp\left( -\int_{x_0}^{x} p(t)\, dt \right).$

Remarks

Proof idea: a computation with the equation gives $W' = -p W$ , a linear first-order equation for $W$ . So $W$ is zero at each point or at no point of $I$ . For an equation of order $n$ with coefficient $a_{n-1}$ of $y^{(n-1)}$ , the same formula holds with $p$ replaced by $a_{n-1}$ . Liouville's formula extends it to linear systems. Niels Henrik Abel found the identity.

Reading path

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Differential equation concept
Function concept
Interval concept
Wronskian definition
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
Homogeneous linear equation definition
Abel's identity statement

Tag	`0VGWKX2JHB`
URI	`m2://0VGWKX2JHB`
Aliases	`abel-formula`
Kind	statement, theorem
Writing	W1 stated
MSC	34A30
Status	This copy of the site has no build data. Run `m2 status 0VGWKX2JHB`.
Uses	Wronskian, Homogeneous linear equation
Same as	Wikidata Q318714 Wikipedia Abel's identity

Abel's identity

Statement

Remarks

Reading path

Further reading

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