Chazy equation

In mathematics, the Chazy equation is the differential equation

$\frac{d^{3}y}{d{x}^3}=2y\frac{d^{2}y}{d{x}^2}-3{\left(\frac{dy}{dx}\right)}^2.$

It was introduced by Jean Chazy (1909, 1911) as an example of a third-order differential equation with a movable singularity that is a natural boundary for its solutions.

One solution is given by the Eisenstein series

$E_2(\tau )=1-24\sum {\sigma }_1(n)q^n=1-24q-72q^2-\cdots .$

Acting on this solution by the group SL₂ gives a 3-parameter family of solutions.