What might be meant by “closed form explicit analytical solution” to an ordinary differential equation?

A certain on-line service claims, "We can find a closed form explicit solution for your ordinary differential equation" (and they supply a price to do so once you submit your ODE to them).

In fact, they claim that "you can find a closed form explicit analytical solution for any ordinary differential equation....a closed form solution expressed in useful functions...."

Of course the claim is manifestly false if they mean in terms of elementary functions. And in the example they provide, they introduce a small perturbation and then provide a solution of the perturbed ODE, not the original one.

What might they mean by their claim?

I didn't want to provide the URL explicitly here, but if you do a web search for

explicit solution for your ordinary differential equation movie clip


then you'll find the web site.

-
Sounds like effective marketing to me. –  copper.hat Oct 3 '12 at 15:51
Since my original post, I looked at the company's page where they give details on pricing. And they charge US $200 more if the client does not supply a numerical solution. So I now believe that all they do is to provide a closed-form analytic function that fits well the numeric solution. – murray Oct 4 '12 at 0:03 I suspect so, for the differential equation$\dot{x}(t) = e^{-\frac{t^2}{2}}\$ doesn't have a closed form solution in terms of elementary functions. –  copper.hat Oct 4 '12 at 0:50
The never actually said "elementary", just "analytic", "closed form", and "useful". –  murray Oct 4 '12 at 17:36
True, but closed form generally implies expressed in terms of a finite number of operations involving 'simple' functions. –  copper.hat Oct 4 '12 at 19:49