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  "body": "\n## Definition\n\nLet $I$ be an open interval and let $q_0, q_1, q_2 : I \\to \\mathbb{R}$ be continuous functions, with $q_0$ and $q_2$ not identically zero.\nA Riccati equation is an [[ordinary differential equation|0P97V20RW7]] of the form\n$$y' = q_0(x) + q_1(x)\\, y + q_2(x)\\, y^2.$$\n\n## Remarks\n\nIf $q_2 = 0$, the equation is [[linear|0Z3YKRQZ48]]. If $q_0 = 0$, it is a [[Bernoulli equation|0ZB84QCDKR]] with $n = 2$.\nIf $y_1$ is one solution, then the substitution $y = y_1 + 1/v$ gives a linear first-order equation for $v$.\nIf $q_2$ has no zero and is differentiable, the substitution $y = -u' / (q_2 u)$ gives the second-order linear equation $u'' - \\left(q_1 + q_2'/q_2\\right) u' + q_0 q_2\\, u = 0$.\nExample: $y' = 1 + y^2$ has the solutions $y = \\tan(x + C)$.\n\n## Explanations\n\n### L1 human\n\nA Riccati equation is the next case after a linear equation: the right side is a quadratic polynomial in $y$.\nIn general, there is no formula for its solutions.\nBut if you know one solution, you can find all the others.\n",
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