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 {{harvs|txt|authorlink=Jean Chazy|first=Jean|last= Chazy|year1=1909|year2=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

{{citation|first=J.|last= Chazy|title= Sur les équations différentielles dont l'intégrale générale est uniforme et admet des singularités essentielles mobiles|journal= C. R. Acad. Sci. Paris |issue=149 |year=1909}}
{{citation|first=J.|last= Chazy|title= Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes|journal=Acta Mathematica |volume=34 |year=1911|pages= 317–385|doi=10.1007/BF02393131|doi-access=free|hdl=2027/mdp.39015080126587|hdl-access=free}}
{{citation|mr=1368067

|title=Symmetry and the Chazy equation

|journal= Journal of Differential Equations|volume=124 |year=1996|issue=1|pages= 225–246|doi=10.1006/jdeq.1996.0008|bibcode=1996JDE...124..225C|doi-access=free}}

Category:Ordinary differential equations