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> It is a more or less pointless statement, but all it means is that 'foo(x)' gives the curve that solves the ODE.

Oh, one more thing: you say, and I understand why, that this is "more or less pointless", but it's not! It is a perfectly good way of producing new functions. For some reason (alphabetical filing in my mind?), I always think first of the Airy function (https://en.wikipedia.org/wiki/Airy_function), but the Bessel functions (https://en.wikipedia.org/wiki/Bessel_function) and, more generally, most (all?) special functions (https://en.wikipedia.org/wiki/Special_functions) also arise in this way.

In fact, even the logarithmic function (boringly, via $\dot y = 1/x$) and the exponential and sine functions (more interestingly, via $\dot y = y$ and $\ddot y = -y$) can be defined this way (by imposing suitable initial conditions, once you know the relevant existence and uniqueness theorems). They can also be defined by their power series, without direct reference to differential equations, but I find such a definition hard to motivate without reference to the differential-equations definition.

Yes, good point. I was actually thinking about special functions and that they are usually defined through differential equations (so, yeah, I more or less stole the idea from there and did not cite it properly :-/ but I did not want to bring all of that in to my comment).

Thanks for elaborating on this.

> A DE "not having a solution" and "not having a solution in terms of elementary functions" are two very different things. "Not having a solution" means (or at least _should_ mean) you could not even numerically solve it in a small neighbourhood (e.g. the curve 'foo(x)' does not exists), "not in terms of elementary functions" means no analytical expression in terms of trigonometric, hyperbolic, exponentials, powers, and so on, can be written down.

I agree with all this; in fact, it's almost exactly what I meant by:

> there is a huge difference between an equation that doesn't have a solution and one whose solution we can't find.

I add the caveat 'almost' because not having a solution is an even stricter condition than not being locally soluble numerically (I think—I'm no expert on numerical methods); but there are equations that meet this stricter condition.

'I say "'foo(x)' is the function which describes the solution to this ODE"'

You can't say this until you've proven that a solution exists (and that it's unique, if you really mean 'the' solution). I think that was part of the point.

Now lets say I give you a non-linear ODE that no-one can solve, but I say "'foo(x)' is the function which describes the solution to this ODE". It is a more or less pointless statement, but all it means is that 'foo(x)' gives the curve that solves the ODE.

Sounds like how we got Bessel functions, no?

Yep; see https://news.ycombinator.com/item?id=9655901 above