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For example, in the differential equation $y' + \frac{y}{x} = 3$, the integrating factor should be $e^{\int \frac1x dx} = e^{\ln|x|} = |x|$, but usually $x$ is used instead.

I investigated it a little and it seems like somehow both $x$ and $|x|$ would work: if we multiply through by $x$, $(xy)' = xy' + y$. But also, if we multiply through by $|x|$, $$(|x|y)' = |x|y' + {\rm sgn}(y)y = |x|y' + \frac{|x|}{x}y.$$

Christian E. Ramirez
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Joe C.
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    Your domain cannot contain both positive and negative $x$-values, and if $\mu$ is an integrating factor, so is $-\mu$ (why?). – Ted Shifrin Feb 27 '23 at 18:28
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    Why can't the domain contain both positive and negative values? – Joe C. Feb 27 '23 at 18:52
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    Because $x=0$ is not in the domain. – Ted Shifrin Feb 27 '23 at 18:53
  • I think that boils down to the same question, but the answer doesn't really explain why, only that both ln(x) and ln|x| happen to work. And I think this question is more focused on the ln(x) vs ln|x| for future readers – Joe C. Feb 27 '23 at 19:20
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    @JoeC. Actually, in that previous answer, $\ln(x)$ does not just happen to work, and the answer does explain how retaining the absolute value symbol then taking cases returns the same solution as just dropping the absolute value symbol. In any case, here's a more explicit (and general) justification: Deciding when to drop the absolute values in differential equation?. Hope it helps! – ryang Feb 27 '23 at 20:51
  • Here's another very similar question: https://math.stackexchange.com/questions/4230736/is-it-necessary-to-consider-absolute-values-when-solving-the-differential-equati – Hans Lundmark Feb 27 '23 at 21:14

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