---
tag: 0VGWKX2JHB
kind: statement
role: theorem
name: Abel's identity
aliases: [abel-formula]
msc: [34A30]
uses: [0N5NF1RGBP, 0EQZSAB9EC]
sources:
  - url: https://en.wikipedia.org/wiki/Abel%27s_identity
    locator: Statement
same_as:
  wikidata: Q318714
  wikipedia: Abel's identity
---

## 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|0N5NF1RGBP]], 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|09PS3S0EF6]] extends it to linear systems.
Niels Henrik Abel found the identity.

## Explanations

### L1 human

You can find the Wronskian of two solutions without knowing the solutions.
Only the coefficient $p$ of $y'$ matters.
For $y'' + y = 0$, we have $p = 0$, so the Wronskian of two solutions is constant.