In mathematics, the Chazy equation is the differential equation
d 3 y d x 3 = 2 y d 2 y d x 2 − 3 ( d y d x ) 2 . {\displaystyle {\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 ( τ ) = 1 − 24 ∑ σ 1 ( n ) q n = 1 − 24 q − 72 q 2 − ⋯ . {\displaystyle 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