Chazy equation

In mathematics, the Chazy equation is the differential equation

\frac{d^3y}{dx^3} = 2y\frac{d^2y}{dx^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.

References

Chazy, J. (1909), "Sur les équations différentielles dont l'intégrale générale est uniforme et admet des singularités essentielles mobiles", C.R. Acad. Sc. Paris (149)
Chazy, J. (1911), "Sur les équations diﬀérentielles du troisième ordre et d’ordre supérieur dont l’intégrale a ses points critiques ﬁxes", Acta Math. 34: 317–385, doi:10.1007/BF02393131
Clarkson, Peter A.; Olver, Peter J. (1996), "Symmetry and the Chazy equation", J. Differential Equations 124 (1): 225–246, doi:10.1006/jdeq.1996.0008, MR 1368067