Chazy equation

In mathematics, the Chazy equation is the differential equation

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

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. Sci. Paris (149)
• Chazy, J. (1911), "Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes", Acta Mathematica, 34: 317–385, doi:10.1007/BF02393131, hdl:2027/mdp.39015080126587
• Clarkson, Peter A.; Olver, Peter J. (1996), "Symmetry and the Chazy equation", Journal of Differential Equations, 124 (1): 225–246, Bibcode:1996JDE...124..225C, doi:10.1006/jdeq.1996.0008, MR 1368067