This is a (perhaps) naive question, but one that I have been thinking about lately. Is it a true statement that all functions (elementary or special) can be defined as the solution to a particular differential equation? That is, any function $f(x)$ can be defined by a solution to
$\quad F(x,y,y',..,y^{(n)}) = 0 $
with (possibly) appropriate boundary conditions. For example, $e^{x}$ can be defined as the solution to:
$y' - y = 0 \quad$ where $\quad y(0)=1$.
A series solution gives you exactly the power series of the exponential function and we use this as its definition. I think the same is true for the trigonometric functions and Bessel functions (and others). Are there any special functions that cannot be defined through the solution of a differential equation?
Thanks.
The function $f(x)=\begin{cases}-1 \text{ if }x<0 \\ 1 \text{ if } x\geq 0 \end{cases}$ can not be the solution of a differential equation, not even if we consider the weak derivative