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  "body": "\n## Definition\n\nLet $D$ be an open subset of $\\mathbb{R}^2$ and let $M, N : D \\to \\mathbb{R}$ be continuous functions.\nThe equation\n$$M(x, y) + N(x, y)\\, y' = 0,$$\nalso written $M\\, dx + N\\, dy = 0$, is exact on $D$ if there is a differentiable function $\\psi : D \\to \\mathbb{R}$ with\n$$\\frac{\\partial \\psi}{\\partial x} = M \\quad \\text{and} \\quad \\frac{\\partial \\psi}{\\partial y} = N \\quad \\text{on } D.$$\nSuch a function $\\psi$ is a potential function of the equation.\n\n## Remarks\n\nLet the equation be exact. Then a differentiable function $y$ on an interval, with graph in $D$, is a [[solution|04SW7WECS8]] if and only if $\\psi(x, y(x))$ is constant. This follows from the chain rule: $\\frac{d}{dx} \\psi(x, y(x)) = M + N y'$.\nSo the solutions are given implicitly by $\\psi(x, y) = C$.\nExample: $2 x y + x^2 y' = 0$ is exact with $\\psi = x^2 y$, and its solutions satisfy $x^2 y = C$.\nThe [[test for exactness|0MSCQNFTM8]] decides whether an equation is exact.\n\n## Explanations\n\n### L1 human\n\nAn exact equation is the total derivative of one function $\\psi$ set to zero.\nSo $\\psi$ stays constant along each solution, and the solution curves are the level curves of $\\psi$.\nTo solve the equation, you find $\\psi$ by integration.\n",
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