A common step in solving many PDEs is to write a multivariable solution function as a product of two more or single variable functions. For example, if given,
$$ \alpha\,\frac{\partial U(x,t)}{\partial x} = \frac{\partial U(x,t)}{\partial t}, $$
one may begin be writing,
$$ U(x,t) = X(x) T(t). $$
My question is, what are the necessary conditions for writing a multi variable function as a product of several single variable functions? Furthermore, are there any cases where this cannot be done?
On
A sufficient and necessary condition is the following.
A function $F:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}:(x,y)\mapsto F(x,y)$ can be written as $F(x,y)=f(x)g(y)$ for some functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ if and only if for every $x,y,w,z\in\mathbb{R}$ it holds that $$F(x,y)F(z,w)=F(x,w)F(z,y).$$
The implication "$\implies$" follows directly from the commutativity of multiplication and the other implication is obtained by first noticing that the zero function $F=0$ can obviously be written as a product of functions of a single variable, and if there exists some $(a,b)\in\mathbb{R}^2$ such that $F(a,b)\neq 0$, then we get that $\forall x,y\in\mathbb{R}$ it holds $$F(x,y)F(a,b)=F(x,b)F(a,y),$$ which implies that
$$F(x,y)=F(x,b)\frac{F(a,y)}{F(a,b)}=f(x)g(y).$$
Of course most two-variable functions $U(x,t)$ cannot be written as $X(x) T(t)$. Those that can are, well, those that happen to be of that form...
The point in the context of PDEs like the heat equation is that you want to find a whole sequence of such separable solutions $U_n(x,t)=X_n(x) T_n(t)$ that you can use in order to write the general solution as a linear combination of these simple solutions: $U(x,t)=\sum_{n=0}^\infty c_n U_n(x,t)$.