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  "body": "\n## Example\n\nThe *Airy equation* is\n$$y'' - z\\, y = 0.$$\nIts coefficients are polynomials, so every point of $\\C$ is an [[ordinary point|QK2J038VB4]], and all solutions are entire functions.\nA [[power series solution|08RT9GDDPQ]] $y = \\sum_{k \\ge 0} a_k z^k$ gives $2 a_2 = 0$ and\n$$(k + 2)(k + 1)\\, a_{k+2} = a_{k-1} \\quad \\text{for } k \\ge 1.$$\nWith $a_0 = 1$ and $a_1 = 0$ one gets $y_1 = 1 + \\frac{z^3}{6} + \\frac{z^6}{180} + \\cdots$.\nWith $a_0 = 0$ and $a_1 = 1$ one gets $y_2 = z + \\frac{z^4}{12} + \\frac{z^7}{504} + \\cdots$.\nThey are linearly independent, since their Wronskian at $0$ is $y_1(0) y_2'(0) - y_1'(0) y_2(0) = 1$.\n\n## Remarks\n\nThe standard solutions are the Airy functions $\\operatorname{Ai}$ and $\\operatorname{Bi}$, which are fixed linear combinations of $y_1$ and $y_2$.\nFor real $x \\to +\\infty$, $\\operatorname{Ai}(x)$ decays like $\\exp\\left( -\\frac{2}{3} x^{3/2} \\right)$ up to a power of $x$, and $\\operatorname{Bi}(x)$ grows like $\\exp\\left( \\frac{2}{3} x^{3/2} \\right)$.\nFor $x \\to -\\infty$, both oscillate with an amplitude that decays like $\\abs{x}^{-1/4}$.\nThe point $\\infty$ is an irregular singular point, and $\\operatorname{Ai}$ is the classical example of the [[Stokes phenomenon|TTGF8KCA8C]].\n\n## Explanations\n\n### L1 human\n\nThe Airy equation is the simplest equation whose solutions switch from oscillating to growing or decaying.\nFor $x \u003c 0$ it behaves like $y'' = -\\abs{x} y$, a spring, so solutions wiggle.\nFor $x \u003e 0$ it behaves like $y'' = x y$, so solutions grow or decay exponentially.\nIt describes light near a rainbow caustic and a quantum particle near a turning point.\n",
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