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  "body": "\n## Definition\n\nLet $I$ be an open interval, and let $p, q : I \\to \\mathbb{R}$ be continuous.\nAn integrating factor of the [[linear first-order equation|0Z3YKRQZ48]] $y' + p(x) y = q(x)$ is a differentiable function $\\mu : I \\to \\mathbb{R}$ with $\\mu(x) \\ne 0$ for each $x \\in I$ and\n$$\\mu'(x) = p(x)\\, \\mu(x) \\quad \\text{for each } x \\in I.$$\nFor such $\\mu$ and each differentiable $y$ on $I$, $\\mu \\left(y' + p y\\right) = (\\mu y)'$.\n\n## Remarks\n\nIf $P$ is an antiderivative of $p$ on $I$, then $\\mu(x) = e^{P(x)}$ is an integrating factor. Each integrating factor is a nonzero constant times this one.\nAfter multiplication by $\\mu$, the equation becomes $(\\mu y)' = \\mu q$. One integration gives the [[solution formula|06B9W45XV0]].\nMore generally, an integrating factor is a function that makes an equation exact after multiplication. See [[integrating factor for a non-exact equation|0CX86CR8JV]].\n\n## Explanations\n\n### L1 human\n\nThe left side $y' + p y$ is not the derivative of a simple expression.\nAfter multiplication by the right function $\\mu$, it becomes the derivative of $\\mu y$, by the product rule.\nThen you can integrate both sides directly.\nFor $y' + y = x$, the factor is $e^x$, and the equation becomes $(e^x y)' = x e^x$.\n",
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