# What Are The Necessary Conditions For a Function to be Separable?

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?

• Pretty sure you could say that if your pde looks like this (just taking two independent variables - the generalization would work as well): $$f_1(x,U,U_x,U_{xx},\cdots)+f_2(x,U,U_x,U_{xx},\cdots)+\cdots+f_n(x,U,U_x,U_{xx},\cdots)+g_1(y,U,U_y,U_{yy},\cdots)+g_2(y,U,U_y,U_{yy},\cdots) +\cdots+g_m(y,U,U_y,U_{yy},\cdots)=0,$$ those would be sufficient conditions. I'm less sure about necessary conditions. You might also need the pde to be linear. – Adrian Keister Sep 25 at 19:10
• There are certainly loads of non-separable pde's out there. – Adrian Keister Sep 25 at 19:12

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: $$U(x,t)=\sum_{n=0}^\infty c_n U_n(x,t)$$.