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  "body": "\n## Summary\n\nLet $M\\, dx + N\\, dy = 0$ be an equation that is not [[exact|0EC7AGJ0PX]] on an open set $D$.\nAn integrating factor for it is a nowhere zero differentiable function $\\mu$ on $D$ such that $\\mu M\\, dx + \\mu N\\, dy = 0$ is exact.\nOn a rectangle, with $M$, $N$ and $\\mu$ of class $C^1$, the [[test for exactness|0MSCQNFTM8]] makes this condition $(\\mu M)_y = (\\mu N)_x$.\nThe two equations have the same solutions, because $\\mu$ has no zero.\n\n## Remarks\n\nIf $(M_y - N_x)/N$ is a function $g(x)$ of $x$ only, then $\\mu(x) = e^{\\int g(x)\\, dx}$ is an integrating factor.\nIf $(N_x - M_y)/M$ is a function $h(y)$ of $y$ only, then $\\mu(y) = e^{\\int h(y)\\, dy}$ is an integrating factor.\nThe [[integrating factor|01THH3Y3YP]] of a linear first-order equation is the first case: $y' + p y - q = 0$ has $M = p y - q$, $N = 1$ and $(M_y - N_x)/N = p$.\n\n## Explanations\n\n### L1 human\n\nSome equations are not exact, but they become exact after multiplication by a good function.\nIn general, it is hard to find this function.\nIn two common cases, the factor depends on one variable only, and a simple integral gives it.\n",
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