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  "body": "\n## Summary\n\nA linear differential operator $P$ is locally solvable at a point if $P u = f$ has a solution near that point for every smooth $f$.\nOperators with constant coefficients are always locally solvable, but Hans Lewy found a first-order operator with polynomial coefficients that is locally solvable at no point.\nThis discovery started the study of necessary and sufficient conditions for local solvability.\nSee the [[definition|DQ0TSVK7ZT]].\n\n## Explanations\n\n### L1 human\n\nFor an ordinary differential equation such as $y' = f(x)$ one can always find a solution near any point, by integrating.\nIt was long expected that every reasonable linear partial differential equation also has local solutions.\nLewy's example shows that this is false: there is a simple first-order equation $L u = f$ in three variables, with smooth $f$, that has no solution in any open set.\nLocal solvability is the property that rules this out.\n",
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