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  "body": "\n## Summary\n\nPainlevé transcendents are solutions of the six Painlevé equations $\\mathrm{P_I}$ to $\\mathrm{P_{VI}}$, nonlinear second order differential equations with the [[Painlevé property|8EV1G2Y6CC]].\nConsider equations $y'' = R(t, y, y')$ with $R$ rational in $y$ and $y'$ that have this property.\nUp to changes of variables, the six equations are the only ones among them that are not solvable by earlier known functions.\nTheir general solutions are new transcendental functions.\nThey play for nonlinear problems the role that classical special functions play for linear ones, for instance in random matrix theory and integrable systems.\n\n## Explanations\n\n### L1 human\n\nSolutions of nonlinear equations usually have moving singularities whose type depends on the initial values, such as branch points.\nPainlevé and his school looked for second order equations whose only moving singularities are poles, so that solutions stay single-valued around them.\nBesides equations solvable by known functions, they found exactly six new families.\nThe simplest is $\\mathrm{P_I}$: $y'' = 6y^2 + t$.\nIts solutions are [[meromorphic functions|0ZZ7ZYDR8W]] on the whole plane that cannot be written in terms of classical functions.\n",
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