Consider the wave equation in one dimension:
$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0.$
The most general solution of this can be written as $F(x-t)+G(x+t)$ for arbitrary functions $F, G$. It is commonly said that this is a consequence of the factorization
$\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}=\left( \frac{\partial }{\partial t}-\frac{\partial}{\partial x}\right)\left( \frac{\partial }{\partial t}+\frac{\partial}{\partial x}\right).$
Is this a general fact?
More precisely, assume that the differential operator $D$ factors into the product of two (commuting) operators $A, B$, that is $D=AB=BA.$ Is it true that $ \{u\in C^{n}\ :\ Du=0\}=\{F+G\ :\ AF=0\ \text{and}\ BG=0\}?$ Here $n$ is the order of $D$ (which is $2$ for $D=\partial^2_t-\partial^2_x$).