# Separation of Variables

As mentioned to the side, this article will discuss everything from the relevance of Stone-Weierstrass to practical aspects of solving differential equations.

For now though, there's just this cute observation about a condition for a nice smooth function to be multiplicatively separable:

A (twice continuously differentiable) function *F(x,y)* (away from areas where it or its derivatives are zero) can be decomposed as *F(x,y)* = *X(x) Y(y)* if and only if it satisfies the differential equation

This obviously holds if the function is separable in this way:

The other way was slightly surprising the first time I noticed it (trying to simplify a GR metric, in fact):

First, rearrange the equation as

And then observe that this takes the form of the *y* derivative of logarithms:

which can easily be integrated to give

for some function *g(x)* coming from the 'constant' of integration (which was with respect to *y*). But now we can rearrange and integrate once more to find