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.[1] The equation reads as[2][3]

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

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.

Solution

Introducing the transformation ${\displaystyle 4By=(A^{2}-4C-z^{2})x^{2}}$ gives

${\displaystyle xz{\frac {dz}{dx}}=A^{2}+AB-4C\pm Bz-z^{2}.}$

Now, the equation is separable, thus

${\displaystyle {\frac {z\,dz}{A^{2}+AB-4C\pm Bz-z^{2}}}={\frac {dx}{x}}.}$

The denominator on the left hand side can be factorized if we solve the roots of the equation ${\displaystyle A^{2}+AB-4C\pm Bz-z^{2}=0}$ and the roots are ${\displaystyle a,\ b=\pm \left[B+{\sqrt {(2A+B)^{2}-16C}}\right]/2}$, therefore

${\displaystyle {\frac {z\,dz}{(z-a)(z-b)}}={\frac {dx}{x}}.}$

If ${\displaystyle a\neq b}$, the solution is

${\displaystyle x{\frac {(z-a)^{a/(a-b)}}{(z-b)^{b/(a-b)}}}=k}$

where ${\displaystyle k}$ is an arbitrary constant. If ${\displaystyle a=b}$, (${\displaystyle (2A+B)^{2}-16C=0}$) then the solution is

${\displaystyle x(z-a)\exp \left[{\frac {a}{a-z}}\right]=k.}$

When one of the roots is zero, the equation reduces to Clairaut's equation and a parabolic solution is obtained in this case, ${\displaystyle A^{2}+AB-4C=0}$ and the solution is

${\displaystyle x(z\pm B)=k,\quad \Rightarrow \quad 4By=-ABx^{2}-(k\pm Bx)^{2}.}$

The above family of parabolas are enveloped by the parabola ${\displaystyle 4By=-ABx^{2}}$, therefore this enveloping parabola is a singular solution.

References

1. Chrystal G., "On the p-discriminant of a Differential Equation of the First order and on Certain Points in the General Theory of Envelopes Connected Therewith.", Trans. Roy. Soc. Edin, Vol. 38, 1896, pp. 803–824.
2. Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
3. Ince, E. L. (1939). Ordinary Differential Equations, London (1927). Google Scholar.