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Now I need help on a different problem that also involves an Integrating Factor of X and Y:

$$6 + 12x^2y^2 + \left(7x^3y + \tfrac xy\right){dy\over dx} = 0$$

The source says the integrating factor is $xy^\tfrac 13$, but I cannot figure out how they got there; suggestions?

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Trial and error. Note that $$ 6 + 12x^2y^2 = t(xy) $$ and $$ 7x^3y + \tfrac xy=x^2\left(7xy+\frac{1}{xy}\right)=s(x^2,xy) $$ So it is quite reasonable to try an integrating factor of the form $\alpha(xy)$.

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If the integrating factor cannot be found by intuition and if the trial and error approach fails, or if we are reluctant to these uncertain methods, see the straightforward method below (even more tiresome) :

enter image description here

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