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  "body": "\n## Example\n\nLet $f_0, f_1, f_2, f_3$ be continuous functions on an open interval $I$, with $f_3$ not identically zero.\nThe *Abel equation of the first kind* is the first-order [[ordinary differential equation|0P97V20RW7]]\n$$y' = f_3(x)\\, y^3 + f_2(x)\\, y^2 + f_1(x)\\, y + f_0(x).$$\nIts right side is a cubic polynomial in $y$.\nWhen $f_3 = 0$ it would be a [[Riccati equation|0E1ZZBHZ3Z]], so the Abel equation is the next case after the Riccati equation.\n\nInstance: $y' = y^3$, with $f_3 = 1$ and $f_0 = f_1 = f_2 = 0$.\nIt is [[separable|0V4E623THY]], and its nonzero solutions are $y = \\pm (C - 2x)^{-1/2}$ for $x \u003c C/2$.\nCheck: for $y = (C - 2x)^{-1/2}$, $y' = -\\tfrac12 (C - 2x)^{-3/2} \\cdot (-2) = (C - 2x)^{-3/2} = y^3$.\nThe solution tends to infinity as $x \\to C/2$, so it exists only on the interval $(-\\infty, C/2)$, which is bounded on the right.\n\n## Explanations\n\n### L1 human\n\nThe Abel equation of the first kind says that the slope of $y$ is a cubic polynomial in $y$ whose coefficients depend on $x$.\nUnlike linear equations and many Riccati equations, it usually has no solution in closed form.\nThe example $y' = y^3$ shows a typical nonlinear effect: solutions can blow up in finite time.\n",
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