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.
[1] The equation reads as[2][3] where
,
{\displaystyle A,\ B,\ C}
are constants, which upon solving for
d x
, gives This equation is a generalization particular cases of Clairaut's equation since it reduces to a form of Clairaut's equation under condition as given below.
Introducing the transformation
gives Now, the equation is separable, thus The denominator on the left hand side can be factorized if we solve the roots of the equation
and the roots are
, the solution is where
is an arbitrary constant.
) then the solution is When one of the roots is zero, the equation reduces to a special-case of Clairaut's equation and a parabolic solution is obtained in this case,
and the solution is The above family of parabolas are enveloped by the parabola
, therefore this enveloping parabola is a singular solution.