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  "body": "\n## Definition\n\nLet $F(x, y, y') = 0$ be a first-order [[ordinary differential equation|0P97V20RW7]], and let $y_s : I \\to \\mathbb{R}$ be a [[solution|04SW7WECS8]] on an open interval $I$.\nThe solution $y_s$ is singular if uniqueness fails at each point of its graph.\nThis means the following. For each $x_0 \\in I$, there is another solution $y$ on an open interval containing $x_0$ with $y(x_0) = y_s(x_0)$ and $y'(x_0) = y_s'(x_0)$. This solution differs from $y_s$ on each neighbourhood of $x_0$.\n\n## Remarks\n\nSome authors use a weaker meaning: a solution at which the [[initial value problem|075VBY4K4X]] fails to have a unique solution at some point.\nA singular solution is often the envelope of a family of solutions. It is not a member of the [[general solution|0SKYFQAFYD]] for any value of the constants.\nExample: for $(y')^2 = 4 y$, the solutions $y = (x - C)^2$ are tangent to the solution $y = 0$, so $y = 0$ is singular. [[Clairaut's equation|0T6D4PE51C]] gives other examples.\nBy the [[uniqueness theorem|025PKW8AMR]], an equation $y' = f(x, y)$ with $f$ Lipschitz in $y$ has no singular solution.\n\n## Explanations\n\n### L1 human\n\nA singular solution is a solution that touches other solutions at each of its points.\nIt often appears when the method of solution divides by an expression that can be zero.\nSo after you solve an equation, check what happens when the divisor is zero.\n",
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