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