I think Mindscrape is misunderstanding the word "integrating factor". In particular, I cannot see what "variation of parameters" could have to do with "integrating factor". He seems to be confusing "integrating factor" with the method of "undetermined coefficients".
An "integrating factor" is a function of the variables which, if you multiply a differential equation by it, makes the equation "exact"- or, in other terms, if you multiply a differential by it, makes the differential and exact differential. To answer your question, bartsdalemc, no, there is no general method of determining whether an integrating factor involves only one or several of the variables. If there were, the problem of solving a general first order differential equation would become trivial- and, believe me, it is not!
Let me elaborate. I think integrating factors are terrible because they are for evaluating, specifically, first order constant coefficient ODEs. Variation of parameters works for any order and nonconstant coefficients, and in general makes more sense. In my opinion, it is much better to learn the general case.
What do you mean variation of parameters has nothing to do with integrating factors? Integrating factors are merely a shortcut for variation of parameters. While you gain a specific formula for the integrating factor, the way to obtain that formula follows a certain algorithm (maybe a term that made you think I was confused?): multiply by the integrating factor µ(x), turn the equation into an exact differential, use the fact that the integrating factor is an exponential, and obtain the solution.
Variation of parameters will also give the solution, but may take longer. Method of undetermined coefficients, if you are good at guessing, will solve it too.
I just don't understand what the original poster is talking about. By the context of the differential equation you know variables are involved.