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  "body": "\n## Definition\n\nConsider an [[ordinary differential equation|0NT64JV96A]] $F(t, x, x', x'', \\varepsilon) = 0$ for $x(t)$, depending on a small parameter $\\varepsilon \u003e 0$.\nThe *method of multiple scales* (two-timing, in its simplest form) proceeds as follows.\n\n1. Introduce the slow time $T = \\varepsilon t$ and seek $x(t) = X(t, T)$, where $X$ is a function of two independent variables.\n2. Replace $\\frac{d}{dt}$ by $\\partial_t + \\varepsilon \\partial_T$, and expand $X = X_0(t, T) + \\varepsilon X_1(t, T) + \\cdots$.\n3. Collect powers of $\\varepsilon$; this gives equations in $t$ for $X_0, X_1, \\ldots$ in which $T$ is a parameter.\n4. Choose the dependence of $X_0, X_1, \\ldots$ on $T$ so that each $X_k$ stays bounded in $t$.\n   This means that the equations have no resonant forcing that produces secular terms.\n\nThe result is an [[asymptotic expansion|Q26QN6GYZX]] intended to be valid for $0 \\le t \\le C / \\varepsilon$.\nMore slow times $\\varepsilon^2 t, \\varepsilon^3 t, \\ldots$ can be added to reach longer time intervals.\n\n## Remarks\n\nExample: $x'' + 2\\varepsilon x' + x = 0$.\nOrder $0$ gives $\\partial_t^2 X_0 + X_0 = 0$, so $X_0 = A(T) \\cos t + B(T) \\sin t$.\nOrder $1$ gives $\\partial_t^2 X_1 + X_1 = -2 \\partial_t \\partial_T X_0 - 2 \\partial_t X_0 = 2(A' + A) \\sin t - 2(B' + B) \\cos t$.\nBounded $X_1$ requires $A' = -A$ and $B' = -B$, so $X_0 = e^{-T} (a \\cos t + b \\sin t)$.\nEvery exact solution has the form $e^{-\\varepsilon t} \\bigl(c_1 \\cos(\\sqrt{1 - \\varepsilon^2}\\, t) + c_2 \\sin(\\sqrt{1 - \\varepsilon^2}\\, t)\\bigr)$.\nIts frequency differs from $1$ by $O(\\varepsilon^2)$, so the approximation has error $O(\\varepsilon)$ for $0 \\le t \\le C / \\varepsilon$.\n\nNon-example: the regular expansion $x = x_0(t) + \\varepsilon x_1(t)$ with one time gives $x_1$ containing $t \\cos t$ and $t \\sin t$ terms, so the error is not small when $t \\ge 1/\\varepsilon$.\n\n## Explanations\n\n### L1 human\n\nUse a fast clock and a slow clock at the same time, and let the slow clock govern how the fast motion changes.\n",
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