Last semester, when setting a question for my first-year calculus class, I was trying to do something that couldn’t be solved by online tools like Wolfram Alpha or Symbolab. In particular, I wanted to get the students to practice verifying that a particular function solves a particular differential equation, but without being able to solve the DE.
It turns out that for special parameter values the third Painlevé transcendant, namely
has an absolutely elementary solution. I really wanted to track this down to its source, and ended up finding this:
To verify is a solution to the given case of Painlevé III above requires no clever tricks, no special knowledge. So I tried to make this a question in my (online, open-book, do-at-home) exam…
It turns out that, at least the way I tried to do it, Wolfram Alpha and Symbolab can’t solve Painlevé transcendents 🙂