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  "body": "\n## Definition\n\nLet $\\Omega$ be an open set in $\\R^n$ and let $m \\ge 1$.\nA *linear partial differential equation* of [[order|E7JDFDPM1X]] $m$ in $\\Omega$ is a [[partial differential equation|GC5BDZ6QBC]] of the following form.\n$$\\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\, \\partial^\\alpha u(x) = f(x), \\quad x \\in \\Omega.$$\nHere the sum runs over [[multi-indices|0PJXBR6QA3]] $\\alpha$.\nThe coefficients $a_\\alpha$ and the right-hand side $f$ are given functions on $\\Omega$, and $a_\\alpha$ is not identically zero for some $\\alpha$ with $\\abs{\\alpha} = m$.\nThe map $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha \\partial^\\alpha$ is the associated *linear differential operator*.\nThe equation is *homogeneous* if $f = 0$ and *inhomogeneous* otherwise.\nIt has *constant coefficients* if every $a_\\alpha$ is constant.\n\n## Remarks\n\nExample: $\\partial_x^2 u + x^2 \\partial_y^2 u + e^y u = \\sin x$ on $\\R^2$ is linear of order $2$, with coefficients $1$, $x^2$ and $e^y$.\nNon-example: $\\partial_x^2 u + u^3 = 0$ is not linear, because the term $u^3$ is not a coefficient times $u$.\nFor a homogeneous linear equation the solutions in $C^m(\\Omega)$ form a vector space.\nFor an inhomogeneous equation, any two solutions differ by a solution of the homogeneous equation.\nIn short notation the equation reads $P u = f$.\n\n## Explanations\n\n### L1 human\n\nA linear PDE is a weighted sum of derivatives of the unknown, with weights that depend only on the point, set equal to a known function.\n",
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