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  "body": "\n## Statement\n\nLet $R(z, w)$ be a [[rational function|0XDK9CGYKJ]] of two complex variables.\nSuppose the [[ordinary differential equation|0NT64JV96A]]\n$$w'(z) = R\\big(z, w(z)\\big)$$\nhas a transcendental [[meromorphic|0ZZ7ZYDR8W]] solution on $\\C$.\nHere transcendental means that the solution is not a rational function.\nThen there are rational functions $a_0, a_1, a_2$ of $z$ such that\n$$R(z, w) = a_0(z) + a_1(z) w + a_2(z) w^2.$$\nSo the equation is a [[Riccati equation|0VT2QK6X6Q]] or a linear equation.\n\n## Remarks\n\nA proof with Nevanlinna theory compares the growth of both sides: if $R$ had degree $d$ in $w$, then $T(r, R(z, w)) = d\\,T(r, w) + O(\\log r)$, while $T(r, w')$ is at most $2T(r, w)$ plus a small error.\nThis bounds the degree, and a study of the poles of $w$ excludes nonpolynomial dependence on $w$.\nExample: $w' = 1 + w^2$ has the transcendental meromorphic solution $w = \\tan z$, and its right side is a polynomial of degree $2$ in $w$, as the theorem requires.\nIn contrast, the solutions of $w' = w^3$ are $w = \\pm (c - 2z)^{-1/2}$ and $w = 0$; they have branch points, as the theorem predicts, so no transcendental meromorphic solution exists.\nThe hypothesis that the solution is transcendental is needed: $w = z$ solves $w' = 1 + (w - z)^3$, whose right side has degree $3$ in $w$.\n\n## Explanations\n\n### L1 human\n\nMost first-order differential equations have solutions with branch points, which are not single-valued in the whole plane.\nTake a first-order equation whose right side is rational.\nMalmquist's theorem says that if it has a non-rational solution that is single-valued with only poles, then it must be a Riccati equation.\nSo, apart from Riccati equations, nonlinear first-order equations with rational right side have no such nice global solutions.\n\n### L2 human\n\nThe theorem is the first-order prototype of the classification behind the Painlevé equations, where the same question is asked for second-order equations.\nIt also has an analogue for difference equations $w(z + 1) = R(z, w(z))$, studied in [[Nevanlinna theory of difference equations|9KAQ1REPE8]]: transcendental meromorphic solutions of finite order force strong restrictions on $R$.\nResults of this kind are called Malmquist-type theorems.\n",
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