The page you linked says that he normally uses Mathematica, and fell back to Wolfram|Alpha for that piece when it started locking up.
While researching this article, I managed to lock up my copy of Mathematica several times on balloon-related differential equations, and subsequently got my IP address banned from Wolfram|Alpha for making too many requests.
Oh, suuuuuure, some kind of magic mathematics company run by a reclusive genius who got a Ph.D. from Cal Tech when he was twenty, and/or college dropout and/or megalomanaic bald silicon valley mogul.
answer for those following along: all of the above :)
While researching this article, I managed to lock up my copy of Mathematica several times on balloon-related differential equations, and subsequently got my IP address banned from Wolfram|Alpha for making too many requests.