I think the Lie techniques have been implemented in MAPLE ODE-solver and probably elsewhere. My experience is MAPLE's ODE solver is very powerful, way stronger than I am. Is it "a decision procedure"? This I do not know. There is a decision procedure for indefinite integration by Risch, and various mathematica etc claim to have implemented it, but I claim they are lying. They pretended to implement it, but not really -- it's too hard and too slow so they had to cheat and lie. I.e. if MAPLE or mathematica fails to integrate something, then I deny that's a proof it cannot be done. So given this disbelief, it is logically forced that I must also disbelieve that these softwares can solve any ODE or by failing prove unsolvable. It would be nice, by the way, if Mathematica were to print out "I have really fully completed the Risch procedure, no kidding, and proven this integral is un-doable" but I have never seen any such system print such a thing out. Anyhow, why don't you just feed in the air-drag ODE system to such packages and see what comes out? [As has now been pointed out, "Ray's" constant of the motion was known in 1860, and just having that does not show me how to solve the whole ball of wax except via leaving it in the form of an integral or by resorting to infinite series of some kind -- I doubt Ray fully solved the system to find position as function of time in closed form, and doubt it is possible to do so. I was initially impressed that this constant of the motion had escaped notice for 300+ years and he just found it, but it appears not and it appears the press got hold of this and overhyped it as "solving 300-year old problem!" while actually it was "nice, but not a full solve which is probably impossible" and much of what he found was already published albeit largely forgotten.]