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  "body": "\n## Summary\n\nKrylov–Bogolyubov averaging is a method for differential equations with a small parameter and a fast periodic dependence on time.\nIt replaces the right-hand side by its average over one period and studies the simpler averaged equation.\nThe averaging theorem says that solutions of the two equations stay close over long time intervals, of length of order $1/\\varepsilon$.\nIt is a basic tool for weakly nonlinear oscillations in mechanics and physics.\nSee the [[definition|NGYZK893VE]].\n\n## Explanations\n\n### L1 human\n\nWhen a quantity changes slowly but is shaken quickly back and forth, the fast shaking mostly cancels out.\nOnly the average effect of each period of shaking matters over a long time.\nFor example, $x' = \\varepsilon x \\sin^2 t$ has the averaged equation $y' = \\frac{1}{2} \\varepsilon y$, because $\\sin^2 t$ has average $\\frac{1}{2}$.\nThe exact solution is $x_0 e^{\\varepsilon (t/2 - \\sin(2t)/4)}$, and it differs from $x_0 e^{\\varepsilon t/2}$ only by the small wobbling factor $e^{-\\varepsilon \\sin(2t)/4}$.\n",
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