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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.

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    $\begingroup$ Any differentiable function $f$ satisfies the differential equation $f' = g$ where $g = f'$, so some clarification is needed to get anything nontrivial: perhaps the differential equation should be of the form $F(x, y, ..., y^{(n)}) = 0$ where $F$ is a rational function. $\endgroup$ – Robert Israel Apr 6 '16 at 15:49
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    $\begingroup$ Pretty sure the Riemann zeta function doesn't satisfy any normal kind of differential equation? Someone correct me if I'm wrong $\endgroup$ – ClassicStyle Apr 6 '16 at 16:03
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    $\begingroup$ Thanks @RobertIsrael Have changed my question accordingly and hope that makes more sense. A quick search seems to suggest the Riemann Zeta function is a counterexample here - at least for an algebraic differential equation $\endgroup$ – AloneAndConfused Apr 6 '16 at 16:11
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    $\begingroup$ It may satisfy some strange DE but it is known that the zeta function and gamma function both do not satisfy algebraic differential equations... $\endgroup$ – ClassicStyle Apr 6 '16 at 16:12
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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

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  • $\begingroup$ Another good point. Perhaps I should restrict to functions that are differentiable on the domain which they are defined (essentially on the real line), though this is clearly a strong restriction! $\endgroup$ – AloneAndConfused Apr 6 '16 at 21:00

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