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 asInce, E. L. (1939). Ordinary Differential Equations, London (1927). Google Scholar.

:$\left(\frac\right)^2 + Ax \frac + By + Cx^2 =0$

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

:$\frac = -\frac x \pm \frac (A^2 x^2 - 4By - 4Cx^2)^.$

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 $4By=(A^2-4C-z^2)x^2$ gives

:$xz\frac = A^2 + AB - 4C \pm Bz - z^2.$

Now, the equation is separable, thus

:$\frac = \frac.$

The denominator on the left hand side can be factorized if we solve the roots of the equation $A^2 + AB - 4C \pm Bz - z^2=0$ and the roots are a,\ b = \pm \left B +\sqrt \right2, therefore

:$\frac = \frac.$

If $a\neq b$, the solution is

:$x \frac = k$

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

:x(z-a) \exp \left frac a \rightk.

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

:$x(z\pm B)=k, \quad \Rightarrow \quad 4By = - AB x^2 - (k\pm Bx)^2.$

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

References

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Equations of physics
Ordinary differential equations