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  "body": "\n## Summary\n\nAn [[ordinary differential equation|0P97V20RW7]] of [[order|0XK0G19SK8]] $n$ is explicit when it has the form $y^{(n)} = f\\left(x, y, y', \\dots, y^{(n-1)}\\right)$ for a given function $f$.\nIt is implicit when it has the general form $F\\left(x, y, y', \\dots, y^{(n)}\\right) = 0$ and is not solved for $y^{(n)}$.\nEach explicit equation is also implicit, with $F = y^{(n)} - f$.\n\n## Remarks\n\nThe equation $y' = x^2 + y^2$ is explicit. The equation $(y')^2 + y^2 = 1$ is implicit.\nIf $\\partial F / \\partial y^{(n)}$ is not zero at a point, the implicit function theorem gives an explicit equation near that point.\nThe basic existence theorems, such as the [[Picard–Lindelöf theorem|0P6KRD5B7A]], are for explicit equations.\n\n## Explanations\n\n### L1 human\n\nAn explicit equation tells you the highest derivative directly, as a formula in the lower ones.\nAn implicit equation only gives a relation, and you may have to solve it for $y'$ first.\nFor $(y')^2 + y^2 = 1$ you get two explicit equations, $y' = \\sqrt{1 - y^2}$ and $y' = -\\sqrt{1 - y^2}$.\n",
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