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  "body": "\n## Definition\n\nLet $I$ be an open interval, let $Q \\colon I \\to \\R$ be smooth with $Q(x) \\neq 0$ for all $x \\in I$, and let $\\varepsilon \u003e 0$ be a small parameter.\nConsider the linear [[ordinary differential equation|0NT64JV96A]]\n$$\\varepsilon^2 y''(x) = Q(x) \\, y(x), \\quad x \\in I.$$\nThe *WKB ansatz* is $y = \\exp\\bigl(\\frac{1}{\\varepsilon} \\sum_{n \\ge 0} \\varepsilon^n S_n(x)\\bigr)$.\nInserting it and collecting powers of $\\varepsilon$ gives $S_0'^2 = Q$ at order $\\varepsilon^0$ and $2 S_0' S_1' + S_0'' = 0$ at order $\\varepsilon^1$.\nThe *WKB approximation*, or *Liouville–Green approximation*, keeps these two orders.\nIt gives the two functions\n$$y_\\pm(x) = Q(x)^{-1/4} \\exp\\Bigl(\\pm \\frac{1}{\\varepsilon} \\int_{x_0}^{x} \\sqrt{Q(s)} \\, ds\\Bigr),$$\nwhere $x_0 \\in I$ and the roots are taken with a fixed branch.\nOn compact subintervals of $I$, the equation has solutions $y = y_\\pm \\, (1 + O(\\varepsilon))$ as $\\varepsilon \\to 0$.\n\n## Remarks\n\nExample: for constant $Q = q \u003e 0$, the formula gives $q^{-1/4} e^{\\pm \\sqrt{q} (x - x_0) / \\varepsilon}$, which solve the equation exactly.\nCheck of the first order: with $S_0' = \\sqrt{Q}$, the second relation gives $S_1' = -S_0'' / (2 S_0') = -Q' / (4 Q)$, so $e^{S_1} = Q^{-1/4}$.\n\nNon-example: the hypothesis $Q \\neq 0$ fails for the Airy-type equation $\\varepsilon^2 y'' = x y$ on an interval containing $0$.\nThe point $x = 0$ is a *turning point*, where $Q^{-1/4}$ blows up; there the WKB formulas are not valid, and solutions are matched through Airy functions.\n\nThe series $\\sum \\varepsilon^n S_n$ is in general only an [[asymptotic expansion|Q26QN6GYZX]], not a convergent series.\nWhen $Q \u003c 0$, one writes $Q = -k^2$, and the two solutions become $k^{-1/2} e^{\\pm i \\int k / \\varepsilon}$, which oscillate.\n\n## Explanations\n\n### L1 human\n\nThe method guesses that the solution is an exponential with a slowly changing exponent.\nThe first equation fixes the phase, and the second fixes how the amplitude changes.\n",
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