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  "body": "\n## Example\n\nLet $k$ and $y_0$ be real numbers.\nThe [[initial value problem|075VBY4K4X]] $y'(t) = k\\, y(t)$, $y(0) = y_0$ has the unique solution\n$$y(t) = y_0\\, e^{k t}, \\qquad t \\in \\mathbb{R}.$$\nIf $k \u003e 0$ and $y_0 \u003e 0$, the solution grows: this is exponential growth. If $k \u003c 0$ and $y_0 \u003e 0$, the solution decreases to $0$: this is exponential decay.\n\n## Remarks\n\nFor $k \u003e 0$, the doubling time is $\\ln 2 / k$: we have $y(t + \\ln 2 / k) = 2\\, y(t)$ for each $t$.\nFor $k \u003c 0$, the half-life is $\\ln 2 / |k|$. Radioactive decay follows this law.\nUniqueness: if $y$ is a solution, then the derivative of $y(t) e^{-k t}$ is $0$, so $y(t) e^{-kt}$ is the constant $y_0$.\n\n## Explanations\n\n### L1 human\n\nThe equation says that the rate of change is proportional to the present amount.\nA population with no limits, or money with continuous interest, grows like this.\nThe larger the amount, the faster it grows, so the growth gets faster and faster.\n",
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