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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $q_1, q_2 \\colon [a, b] \\to \\R$ be continuous with $q_2(x) \\ge q_1(x)$ for all $x \\in [a, b]$.\nLet $y$ be a solution of $y'' + q_1(x) y = 0$ that is not identically zero and satisfies $y(a) = y(b) = 0$.\nLet $z$ be a solution of $z'' + q_2(x) z = 0$.\nThen $z$ has a zero in the open interval $(a, b)$, unless $q_1 = q_2$ on $[a, b]$ and $z$ is a constant multiple of $y$.\n\n## Remarks\n\nThe proof uses the function $W = y' z - y z'$, whose derivative is $(q_2 - q_1) y z$.\nIf $z$ had no zero in $(a, b)$, one may take $y \u003e 0$ and $z \u003e 0$ on $(a, b)$.\nIntegrating $W'$ over $[a, b]$ gives $y'(b) z(b) - y'(a) z(a) \\ge 0$, while the signs $y'(a) \u003e 0$, $y'(b) \u003c 0$ and $z \\ge 0$ force the left side to be $\\le 0$.\nAnalysing the case of equality gives $q_1 = q_2$ and $z(a) = z(b) = 0$, so $z$ is a multiple of $y$.\nExample: $y = \\sin x$ solves $y'' + y = 0$ with zeros $0$ and $\\pi$; every solution of $z'' + 4 z = 0$, such as $\\cos 2x$, has a zero in $(0, \\pi)$.\nThe inequality $q_2 \\ge q_1$ is needed: with $q_2 = 0$, the solution $z = 1$ of $z'' = 0$ has no zero at all.\nMauro Picone extended the theorem to equations $(p_i y')' + q_i y = 0$ with $p_1 \\ge p_2 \u003e 0$ and $q_2 \\ge q_1$; this is the Sturm–Picone comparison theorem.\n\n## Explanations\n\n### L1 human\n\nA larger coefficient $q$ in $y'' + q y = 0$ pulls the solution back to the axis harder, so it oscillates faster.\nThe theorem makes this precise: between two zeros of a solution of the slower equation, every solution of the faster equation has a zero.\n\n### L2 human\n\nThe theorem lets one bound the zeros of solutions by comparison with equations with constant coefficients.\nIf $m^2 \\le q(x) \\le M^2$ with $m, M \u003e 0$, then the distance between consecutive zeros of a nonzero solution of $y'' + q y = 0$ lies between $\\pi / M$ and $\\pi / m$.\nIt is the main tool of [[oscillation theory|ZH2S50NSZ6]] and it underlies the count of zeros of eigenfunctions in Sturm–Liouville theory.\nFor equations of the form $y'' + q y = 0$, the case $q_1 = q_2$ gives the [[Sturm separation theorem|YEBPTXZ1FS]].\n",
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