# Finding an integrating factor, function of X and Y.

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?

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 $\alpha(xy)$.