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## Homework Statement

I'm reading a chapter out of Elem. DE 6th Edition by Rainville and Bedient (Ch 4 pg 61) titled integrating factors found by inspection. To explain it, the authors start with an equation, which is grouped to become:

[itex]y dx + x dy + x^3y^2 dy = 0[/itex]

which then becomes:

[itex]\frac{d(xy)}{(xy)^3} + \frac{dy}{y} = 0[/itex]

I am unsure of how they get from there to the next step which is this....

[itex]-\frac{1}{2x^2y^2} + ln|y| = -ln|c|[/itex]

## The Attempt at a Solution

Trying to figure out the intermediate steps I start by doing this...

[itex]\frac{d(xy)}{x^3y^2} = - \frac{dy}{y}[/itex]

And then I'm guessing they integrate both sides. However, if that's the case, what would I be integrating the left side with respect to? I expected the equation to become this...

[itex]\frac{1}{x^2y^2} + ln|y| = c[/itex]

Thank you.

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