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  "body": "\n## Example\n\nLet $x_0 \u003e 0$ and consider the [[initial value problem|075VBY4K4X]] $x'(t) = x(t)^2$, $x(0) = x_0$.\n[[Separation of variables|0V4E623THY]] gives the solution\n$$x(t) = \\frac{x_0}{1 - x_0 t}, \\qquad t \\in \\left(-\\infty, \\frac{1}{x_0}\\right).$$\nThis is the [[maximal solution|019VSXSFMQ]]: $x(t) \\to \\infty$ as $t \\to 1/x_0$ from below.\nSo the solution blows up at the finite time $1/x_0$, although the field $f(x) = x^2$ is smooth on all of $\\mathbb{R}$.\n\n## Remarks\n\nThe blow-up time $1/x_0$ depends on the initial value. Larger initial values blow up sooner.\nFor $x_0 \u003c 0$, the maximal interval is $(1/x_0, \\infty)$, and for $x_0 = 0$ the solution is $0$ on $\\mathbb{R}$.\nThe field $x^2$ is not globally Lipschitz. A [[globally Lipschitz field|0BFD3A6CRH]] has no blow-up.\nNear the blow-up time, $x(t)$ behaves like $(1/x_0 - t)^{-1}$. This is a finite-time singularity.\n\n## Explanations\n\n### L1 human\n\nThe rate of growth is the square of the size, so the growth feeds itself very fast.\nThe solution reaches infinity in finite time, unlike exponential growth.\n",
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        "url": "https://en.wikipedia.org/wiki/Ordinary_differential_equation"
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