Post History

I don't think there's a satisfying answer to this question currently. The very first problem – which is probably surprising – is that there isn't a widely accepted, general definition of "separation of variables".

An obvious approach to studying separation of variables would be to apply the techniques of Lie theory. This is what Miller does in "Symmetry and Separation of Variables" which is cited in the Wikipedia article. A quick look at the [table of contents](https://assets.cambridge.org/97805211/77399/frontmatter/9780521177399_frontmatter.pdf), however, does not exactly suggest a universal theory. In [his book review](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-1/issue-6/Review-Willard-Miller-Jr-Symmetry-and-separation-of-variables/bams/1183544920.full), T. Koornwinder complained at the lack of definition of "separation of variables" in the book. He followed up by writing [A Precise Definition of Separation of Variables](https://www.semanticscholar.org/paper/A-precise-definition-of-separation-of-variables-Koornwinder/47f5bc8e1471344ca8cf258742e7dc26fdfd08f1). As far as I can tell, his definition is not widely adopted. I don't know that is due to any issue with his definition, or if mathematicians just don't care that much about a precise definition.

That said, the article states that it will give necessary and sufficient conditions for when separation of variables (in his sense) can be applied for linear homogeneous second-order partial differential equations which may be of interest to you. Unfortunately, when I was recently looking for what Lie theory says about separation of variables, Miller's book still seems to be the main reference.

While you could use Koornwinder's conditions as an answer to your question of knowing when an equation is separable for a common class of PDEs, in practice, it's probably easier to simply assume a separable ansatz and see if you can separate the variables.

As to why it works, it's pretty simple: you are simply making some assumptions about the form of the solution. It may be that no solutions satisfy those assumptions, in which case you won't find any. For example, you could just as well have assumed $u(x,t)=X(x)$ and that would have worked, though it would give you a less complete set of solutions. On the other hand, you could have assumed $u(x,t)=T(t)$ and you would have found no solutions unless $f(x)$ was identically $0$.

To answer another of your questions, the assumption that the solution is separable does exclude other non-separable solutions. For example, see [On the Production of Nonseparable Solutions of Linear Partial Differential Equations](https://core.ac.uk/download/pdf/322470896.pdf) and its citations. So the technique doesn't, in general, find all solutions. You would need to verify that the same way you would verify that you have all solutions to a differential equation no matter how you arrived at them. This is, of course, itself often not an easy problem.