{
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  "body": "\n## Statement\n\nLet $I$ be an open interval, let $p, q : I \\to \\mathbb{R}$ be continuous, and let $y_1, y_2$ be solutions on $I$ of\n$$y'' + p(x)\\, y' + q(x)\\, y = 0.$$\nLet $W = y_1 y_2' - y_1' y_2$ be their [[Wronskian|0N5NF1RGBP]], and let $x_0 \\in I$.\nThen for each $x \\in I$,\n$$W(x) = W(x_0) \\exp\\left( -\\int_{x_0}^{x} p(t)\\, dt \\right).$$\n\n## Remarks\n\nProof idea: a computation with the equation gives $W' = -p W$, a linear first-order equation for $W$.\nSo $W$ is zero at each point or at no point of $I$.\nFor 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.\nNiels Henrik Abel found the identity.\n\n## Explanations\n\n### L1 human\n\nYou can find the Wronskian of two solutions without knowing the solutions.\nOnly the coefficient $p$ of $y'$ matters.\nFor $y'' + y = 0$, we have $p = 0$, so the Wronskian of two solutions is constant.\n",
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    "name": "Abel's identity",
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  "name": "Abel's identity",
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