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Chrystal's equation

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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 are constants, which upon solving for , 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.

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Solution

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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 , therefore

If , the solution is

where is an arbitrary constant. If , () 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.

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References

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