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  "body": "\n## Statement\n\nLet $D \\subseteq \\R^2$ be open and let $f : D \\to \\R$ be continuous and locally Lipschitz in its second variable.\nLet $a \u003c b$, let $u : [a, b] \\to \\R$ be a [[solution|04SW7WECS8]] of $u' = f(t, u)$, and let $v : [a, b] \\to \\R$ be differentiable, with the graphs of $u$ and $v$ in $D$.\nAssume that $v'(t) \\le f(t, v(t))$ for each $t \\in [a, b]$ and that $v(a) \\le u(a)$.\nThen $v(t) \\le u(t)$ for each $t \\in [a, b]$.\n\n## Remarks\n\nProof idea: suppose $v(t_1) \u003e u(t_1)$ and let $s$ be the last time before $t_1$ with $v(s) \\le u(s)$, so $v(s) = u(s)$.\nJust after $s$ the difference $w = v - u$ is positive and satisfies $w' \\le f(t, v) - f(t, u) \\le K w$ by the [[Lipschitz condition|055VVSN00V]].\nThen $w(t) e^{-K(t - s)}$ is non-increasing and $w(s) = 0$, so $w \\le 0$ there, a contradiction; this is the differential form of [[Grönwall's inequality|07EJHQYR9A]].\nExample: $v(t) = 0$ satisfies $v' \\le 1 + v^2$, so the solution $u = \\tan t$ of $u' = 1 + u^2$, $u(0) = 0$, is $\\ge 0$ on $[0, \\pi/2)$; with $v(t) = t$, which satisfies $v' = 1 \\le 1 + v^2$, it gives $\\tan t \\ge t$ there.\nThe Lipschitz hypothesis is needed: for $f(t, x) = 2\\sqrt{\\abs{x}}$, the solution $u = 0$ and the function $v = t^2$ satisfy $v' = 2\\sqrt{v}$ and $v(0) = u(0)$, but $v \u003e u$ for $t \u003e 0$.\nWithout Lipschitz conditions the conclusion holds when $u$ is replaced by the maximal solution of the initial value problem.\n\n## Explanations\n\n### L1 human\n\nIf a quantity always grows at most as fast as the rule $u' = f(t, u)$ allows, and starts no higher, then it stays below the solution of that rule.\nOne can bound solutions of a hard equation by solutions of an easier one.\n\n### L2 human\n\nComparison arguments give a priori bounds, blow-up times and global existence results for scalar equations.\nFor systems, one applies the theorem to a scalar quantity such as $\\norm{x(t)}^2$ or a [[Lyapunov function|04P7B5XC22]].\nThe [[Bihari–LaSalle inequality|9KMJSKNZ72]] and Grönwall's inequality are comparison results in integral form.\nThe [[Sturm comparison theorem|W7VXNYZQN2]] is a different result, about zeros of solutions of second-order linear equations.\n",
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