TITLE: does the j-invariant satisfy a rational differential equation?
QUESTION [14 upvotes]: Let $j$ be the Klein $j$-invariant (from the theory of modular functions).
Does $j$ satisfy a differential equation of the form $j^\prime (z) = f(j(z),z)$ for 
any rational function $f$?

REPLY [2 votes]: A third-order differential equation for $j(\tau)$ is gotten via the Schwarzian derivative. The result is (1.13) of the paper by Harnad: http://arxiv.org/abs/solv-int/9902013