Finding an integrating factor (Differential Equations, exact equations) The given formula is:
$4\left(\frac{x^3}{y^2} + \frac{3}{y} \right)dx + 3 \left(\frac{x}{y^2} + 4y\right)dy = 0$
This gives $M_y = - \frac{8x^3}{y^3} - \frac{12}{y^2}$ and $N_x = \frac{3}{y^2}$.  So clearly as given this equation isn't exact.
The problem text states that I am to "Find an integrating factor and solve the given equation."  However, I cannot find μ using any technique that I know or that I could come up with.
The only ways that the book up to this point has discussed coming up with integrating factors are in the form of $\frac{d\mu}{dx} = \frac{M_y - N_x}{N}\mu$ and $\frac{d\mu}{dy} = \frac{N_x - M_y}{M}\mu$ and the one oddball one where μ is a function of xy: $\frac{d\mu}{dxy} = \frac{N_x - M_y}{xM - yN}\mu$.  Working these and also deriving and working μ(y/x) and μ(x/y).
I tried the final three because on the next problem there's a hint that pulls back to a different problem that was to provide a proof of a case where μ is a function of xy.  And as this is an even-numbered problem, it's trickier than the odd numbered ones.
In the interest of showing my work.  Here are the cases:
$\frac{d\mu}{dx} = \frac{- \frac{8x^3}{y^3} - \frac{15}{y^2}}{\frac{3x}{y^2} + 12y}\mu = Not\, promising...$
$\frac{d\mu}{dy} = \frac{\frac{8x^3}{y^3} + \frac{15}{y^2}}{\frac{4x^3}{y^2} + \frac{12}{y}}\mu = Not\, promising...$
$\frac{d\mu}{dxy} = \frac{- \frac{8x^3}{y^3} - \frac{15}{y^2}}{\frac{4x^4}{y^2} + \frac{15x}{y} + 12y^2}\mu = Really\,not\,promising...$
And for the division ones (are these right?):
$\frac{\partial\mu(\frac{x}{y})}{\partial x} = \frac{\frac{d\mu}{dx}}{y}$
$\frac{\partial\mu(\frac{x}{y})}{\partial y} = -\frac{x\frac{d\mu}{dx}}{y^2}$
And vica versa for the inverted case (are these right?):
$\frac{\partial\mu(\frac{y}{x})}{\partial x} =  -\frac{y\frac{d\mu}{dx}}{x^2}$
$\frac{\partial\mu(\frac{y}{x})}{\partial y} = \frac{\frac{d\mu}{dx}}{x}$
And you can easily verify that nothing good comes of using these ideas.
So... I'm lost.  I probably made a really basic mistake early on, that's causing the numbers to not line up correctly, but for the life of me I just cannot find it.
μ(y) case looks the closest to being good. But it's just not working out.
 A: Is the book you are referring to by any chance this one here? I've found that excerpt - after searching for longer than I care to admit ;) - and your description of the next problem (no. 31) referring back to another problem fits exactly!
Anyway, if you look closely at problem no. 30 (yours), you see that you have misplaced a parenthesis (already in your reddit post - yeah, I found that one, too) which makes this problem way harder than it really is.${}^{[1]}$
The correct inexact differential is
$$
\left(4\frac{x^3}{y^2}+\frac{3}{y}\right)\text{d}x + \left(3\frac{x}{y^2}+4y\right)\text{d}y
$$
so you see that the factors of $3$ and $4$ should be inside the parentheses. Now if you multiply through by $\mu(y)=y^2$, you get
$$
\left(4x^3+3y\right)\text{d}x + \left(3x+4y^3\right)\text{d}y
$$
which gives you an exact differential, since
$$
\partial_y\left(4x^3+3y\right)=3=\partial_x\left(3x+4y^3\right).
$$
In the end, your intuition that the $\mu=\mu(y)$ looked the best was right, the factors were simply slightly off.

${}^{[1]}$ To be fair, I have no idea how to find the integrating factor for the differential as it is posed in the current version (v2). While one immediately sees by comparison with the other problems in the linked PDF that this cannot be the intended procedure, it should be doable in general by using the method of characteristics to solve the semilinear partial differential equation for $\mu(x,y)$, but I've been running in circles on that path for a while now and am out of good ideas.
