Elementary solution to Painlevé III

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

y'' = \frac{1}{y}\left( (y')^2 - BA^2/3 \right) + \frac{1}{x}(By^2 - y'),

has an absolutely elementary solution. I really wanted to track this down to its source, and ended up finding this:

N. A. Lukashevich, “On the theory of Painlevé’s third equation”, Differ. Uravn., 3:11 (1967), 1913–1923 Math-Net.Ru

To verify y = A\sqrt[3]{z} 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…

Not the final question. The parameters A and B were meant to be randomised.

It turns out that, at least the way I tried to do it, Wolfram Alpha and Symbolab can’t solve Painlevé transcendents 🙂

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