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  "body": "\n## Definition\n\nLet $n \\ge 0$ be a real number, called the polytropic index.\nThe *Lane–Emden equation* of index $n$ is the [[ordinary differential equation|0NT64JV96A]]\n$$\\frac{1}{\\xi^2} \\frac{d}{d\\xi} \\left( \\xi^2 \\frac{d\\theta}{d\\xi} \\right) + \\theta^n = 0$$\nfor a function $\\theta$ of $\\xi \u003e 0$, with the initial conditions $\\theta(0) = 1$ and $\\theta'(0) = 0$.\nWhen $n$ is not an integer, the equation is considered on the interval where $\\theta \\ge 0$.\n\n## Remarks\n\nFor a radial function $u(x) = \\theta(\\abs{x})$ on $\\R^3$, the first term is the [[Laplacian|0E0B6RA8K0]] $\\Delta u$, so the equation is $\\Delta u + u^n = 0$.\nExample: for $n = 0$, $\\theta(\\xi) = 1 - \\xi^2/6$ is the solution.\nIndeed, $\\xi^2 \\theta' = -\\xi^3/3$, whose derivative divided by $\\xi^2$ is $-1$, and $\\theta^0 = 1$.\nNon-example: for $n = 1$, the constant $\\theta = 1$ is not a solution, since the left side equals $1$.\nClosed-form solutions are known for $n = 0$, $n = 1$, where $\\theta = \\sin \\xi / \\xi$, and $n = 5$, where $\\theta = (1 + \\xi^2/3)^{-1/2}$.\nIn PDE the name Lane–Emden equation, or Lane–Emden–Fowler equation, is also used for $-\\Delta u = \\abs{u}^{p-1} u$ on a domain of $\\R^N$ with $p \u003e 1$.\n",
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