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  "body": "\n## Example\n\nLet $f$ be a twice continuously differentiable function on an open interval $J$.\nClairaut's equation is the implicit first-order equation\n$$y = x\\, y' + f(y'),$$\nfor an unknown $y$ with $y' \\in J$.\nFor each constant $C \\in J$, the line $y = C x + f(C)$ is a solution. These lines form the [[general solution|0SKYFQAFYD]].\nIf $f'' \\ne 0$ on $J$, the curve with the parametric form $x = -f'(p)$, $y = f(p) - p f'(p)$ for $p \\in J$ is also a solution. It is the envelope of the lines, and it is a [[singular solution|0CZA0NCFWP]].\n\n## Remarks\n\nDifferentiation of the equation gives $y''\\left(x + f'(y')\\right) = 0$. The case $y'' = 0$ gives the lines, and the case $x + f'(y') = 0$ gives the envelope.\nExample: for $f(p) = p^2$, the lines are $y = C x + C^2$, and the singular solution is the parabola $y = -x^2/4$.\nEach line of the general solution is tangent to the singular solution at one point.\n\n## Explanations\n\n### L1 human\n\nClairaut's equation has a family of straight lines as solutions, one for each slope $C$.\nThese lines wrap around a curve, their envelope, and this curve is also a solution.\nNo choice of $C$ gives the envelope, so it is a new, singular solution.\n",
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