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  "body": "\n## Definition\n\nLet $\\delta, K, \\varepsilon$ be real numbers.\nThe Grönwall bound with these parameters is the function $G_{\\delta, K, \\varepsilon} : \\mathbb{R} \\to \\mathbb{R}$ given by\n$$G_{\\delta, K, \\varepsilon}(x) = \\begin{cases} \\delta\\, e^{K x} + \\dfrac{\\varepsilon}{K} \\left( e^{K x} - 1 \\right) \u0026 \\text{if } K \\neq 0, \\\\ \\delta + \\varepsilon x \u0026 \\text{if } K = 0. \\end{cases}$$\n\n## Remarks\n\nThe function $v = G_{\\delta, K, \\varepsilon}$ is the unique solution of $v' = K v + \\varepsilon$, $v(0) = \\delta$.\nFor $\\varepsilon = 0$, it is $\\delta e^{K x}$.\nA form of [[Grönwall's inequality|07EJHQYR9A]] uses it: let $f : [a, b] \\to \\mathbb{R}^n$ be continuous and differentiable on $(a, b)$, with $\\|f(a)\\| \\le \\delta$ and $\\|f'(x)\\| \\le K \\|f(x)\\| + \\varepsilon$ on $(a, b)$. Then $\\|f(x)\\| \\le G_{\\delta, K, \\varepsilon}(x - a)$ on $[a, b]$.\nThe [[estimate for approximate solutions|0JTPMCCZT2]] is stated with this bound.\n\n## Explanations\n\n### L1 human\n\nSuppose that a quantity starts at most at $\\delta$ and grows at most at rate $K$ times itself plus $\\varepsilon$. The Grönwall bound is the largest value that it can reach.\nIt is the solution of the extreme case, where both inequalities are equalities.\n",
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