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  "body": "\n## Definition\n\nLet $I$ be an open [[interval|0Q1D85B8G6]] and let $p, q : I \\to \\mathbb{R}$ be continuous functions.\nThe linear first-order equation with coefficient $p$ and right side $q$ is the equation\n$$y' + p(x)\\, y = q(x)$$\nfor an unknown differentiable function $y : I \\to \\mathbb{R}$.\nIt is homogeneous if $q = 0$ on $I$, and inhomogeneous otherwise.\n\n## Remarks\n\nThe [[solution formula|06B9W45XV0]] gives each solution explicitly, with an [[integrating factor|01THH3Y3YP]].\nThe homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$, where $P$ is an antiderivative of $p$ and $C$ is a constant.\nMany authors write the equation as $y' = a(x) y + b(x)$, with $a = -p$ and $b = q$.\n\n## Explanations\n\n### L1 human\n\nThe equation is linear because $y$ and $y'$ appear only to the first power, with coefficients that depend on $x$.\nThere is no product such as $y\\, y'$ and no function such as $\\sin y$.\nThis structure is the reason why there is a formula for all the solutions.\n",
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