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  "body": "\n## Definition\n\nLet $\\alpha, \\beta, \\gamma, \\delta$ be complex constants, and let $y$ be a function of a complex variable $t$.\nThe six *Painlevé equations* are the [[ordinary differential equations|0NT64JV96A]]:\n\n1. $\\mathrm{P_I}$: $y'' = 6y^2 + t$;\n2. $\\mathrm{P_{II}}$: $y'' = 2y^3 + ty + \\alpha$;\n3. $\\mathrm{P_{III}}$: $y'' = \\frac{(y')^2}{y} - \\frac{y'}{t} + \\frac{\\alpha y^2 + \\beta}{t} + \\gamma y^3 + \\frac{\\delta}{y}$;\n4. $\\mathrm{P_{IV}}$: $y'' = \\frac{(y')^2}{2y} + \\frac32 y^3 + 4ty^2 + 2(t^2 - \\alpha) y + \\frac{\\beta}{y}$;\n5. $\\mathrm{P_V}$: $y'' = \\left( \\frac{1}{2y} + \\frac{1}{y - 1} \\right) (y')^2 - \\frac{y'}{t} + \\frac{(y - 1)^2}{t^2} \\left( \\alpha y + \\frac{\\beta}{y} \\right) + \\frac{\\gamma y}{t} + \\frac{\\delta y (y + 1)}{y - 1}$;\n6. $\\mathrm{P_{VI}}$: $y'' = \\frac12 \\left( \\frac1y + \\frac{1}{y - 1} + \\frac{1}{y - t} \\right) (y')^2 - \\left( \\frac1t + \\frac{1}{t - 1} + \\frac{1}{y - t} \\right) y' + \\frac{y (y - 1)(y - t)}{t^2 (t - 1)^2} \\left( \\alpha + \\frac{\\beta t}{y^2} + \\frac{\\gamma (t - 1)}{(y - 1)^2} + \\frac{\\delta t (t - 1)}{(y - t)^2} \\right)$.\n\nA *Painlevé transcendent* is a solution of one of these six equations.\n\n## Remarks\n\nEach equation has the [[Painlevé property|8EV1G2Y6CC]]: the only movable singularities of its solutions are poles.\nExample: for $\\alpha = 1$, the function $y = -\\frac1t$ solves $\\mathrm{P_{II}}$.\nCheck: $y'' = -\\frac{2}{t^3}$, and $2y^3 + ty + 1 = -\\frac{2}{t^3} - 1 + 1 = -\\frac{2}{t^3}$.\nThis solution is rational, so for special parameters some Painlevé transcendents are classical functions; the generic solution is not.\nNon-example: a solution of $y'' = 6y^2$ is not a Painlevé transcendent, since this equation is not one of the six; its nonconstant solutions are translates of the Weierstrass function $\\wp(t; 0, g_3)$.\nEach of $\\mathrm{P_I}$ to $\\mathrm{P_V}$ is a limit of $\\mathrm{P_{VI}}$ under a suitable coalescence of singular points and parameters.\n",
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