We have an ODE of the form $$M(x,y)\,dx + N(x,y)\,dy = 0$$ in which both $M$ and $N$ are homogeneous functions of the same degree. The standard method for handling such an equation (assuming it's not separable or linear, in which case(s) we have a choice of "standard methods") is to use the substitution $\{y=ux, dy=udx+xdu\}$, obtaining a separable equation. If the resulting integral in 'u' fails to be nice (the one on the 'x' side is always nice), we can try swapping the roles of $x$ and $y$ for an alternative substitution. This is all very standard, AFAIK.
Now, from an exercise in a book (Boyce & DiPrima), I see that any homogeneous equation may be made exact via the integrating factor $$\mu(x,y)=\frac{1}{xM(x,y)+yN(x,y)}.$$ This is great, it illustrates that integrating factors can exist that aren't constant in either variable, and doing it this way is an interesting exercise, as is proving that the given $\mu$ actually works.
However... is there any example of a homogeneous DE where this is a better method than just using the normal substitutions? I haven't encountered an example that makes me think: "This is the nice way to do it!"
Can anyone display a nice example, that plays to the strengths of this technique? Does this method perhaps exist more as a curiosity than as a practical tool? Does the resulting exact equation give us access to some kind of insight we would not have encountered the other way? Is the interesting part simply the fact that any homogeneous equation can be made exact?