Chrystal's equation

In mathematics, Chrystal's equation is a first order nonlinear ordinary differential equation, named after the mathematician George Chrystal, who discussed the singular solution of this equation in 1896. The equation reads as

${\displaystyle \left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0}$

where ${\displaystyle A,\ B,\ C}$ are constants, which upon solving for ${\displaystyle dy/dx}$, gives

${\displaystyle {\frac {dy}{dx}}=-{\frac {A}{2}}x\pm {\frac {1}{2}}(A^{2}x^{2}-4By-4Cx^{2})^{1/2}.}$

This equation is a generalization of Clairaut's equation since it reduces to Clairaut's equation under certain condition as given below.