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  "body": "\n## Summary\n\nA differentially closed field is a [[differential field|GA5PTQGC6G]] in which every differential equation that can have a solution in some extension already has one.\nIt is the analogue, for differential equations, of an [[algebraically closed field|0PF6TM8TWT]] for polynomial equations.\nIn characteristic $0$, the differentially closed fields form an elementary class whose theory, called $\\mathrm{DCF}_0$, is a central example in [[model theory|HFTBG5PMXC]].\n\n## Explanations\n\n### L1 human\n\nIn an algebraically closed field, every nonconstant polynomial has a root.\nA differentially closed field asks the same of equations that involve an unknown $y$ and its derivatives $y', y'', \\dots$.\nSuch fields are huge and abstract: no field of familiar functions, such as $\\C(x)$, is differentially closed.\nFor instance, $\\C(x)$ contains no solution of $y' = 1/x$, since $\\log x$ is not a rational function.\nEvery differential field of characteristic $0$ sits inside a differentially closed one, much as every field sits inside an algebraically closed one.\n",
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