Why is the method of separation of variables so widely used in physics? Why is the method of separation of variables is the only method used in physics for solving partial differential equations despite that they are not the most general solutions? Do these solutions form some kind of complete set? Is there a proof? 
 A: Because this method will work. It reduces many partial differential equations down to  ordinary differential equations and if we can solve those then we’re in business and the method will allow us to get a solution to the partial differential equations.
A: It's definitely not the only method. Separation of variables is convenient to use if the domain is finite, i.e. $x\in[0,1]$.
If the domain is inifnite, i.e. $x\in[0,\infty)$, the fundamental solution is often used instead.
A: Fourier's separation of variables technique was one of the earliest methods for solving linear partial differential equations. A good number of classical problems such as the Laplace equation, the Heat equation, and the wave equation can be solved this way, and the solutions are explicit. (By the way, $x+y = x\cdot 1 + 1\cdot y$ is a separated function.) There are not a lot of other techniques that can be used to find explicit solutions; so the technique remains useful for gaining theoretical and practical information about these classical equations of Engineering and Physics.
A: It is also, in connection with PDFs (Probability Density Functions) a reason of independence for example in the simplest case where the variables are $x$ (space) and $t$ (time) when we know that there is no connection between space and time. 
Indeed a PDF $f(x,t)$ can be written $f_1(x)f_2(t)$ if and only if the associated Random Variables $X$ and $T$ are independent. And this is frequently the case.
