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]
- [math]\displaystyle{ \left(\frac{dy}{dx}\right)^2 + Ax \frac{dy}{dx} + By + Cx^2 =0 }[/math]
where [math]\displaystyle{ A,\ B, \ C }[/math] are constants, which upon solving for [math]\displaystyle{ dy/dx }[/math], gives
- [math]\displaystyle{ \frac{dy}{dx} = -\frac{A}{2} x \pm \frac{1}{2} (A^2 x^2 - 4By - 4Cx^2)^{1/2}. }[/math]
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 [math]\displaystyle{ 4By=(A^2-4C-z^2)x^2 }[/math] gives
- [math]\displaystyle{ xz\frac{dz}{dx} = A^2 + AB - 4C \pm Bz - z^2. }[/math]
Now, the equation is separable, thus
- [math]\displaystyle{ \frac{z \, dz}{A^2 + AB - 4C \pm Bz - z^2} = \frac{dx}{x}. }[/math]
The denominator on the left hand side can be factorized if we solve the roots of the equation [math]\displaystyle{ A^2 + AB - 4C \pm Bz - z^2=0 }[/math] and the roots are [math]\displaystyle{ a,\ b = \pm \left[ B +\sqrt{(2A+B)^2 - 16C} \right]/2 }[/math], therefore
- [math]\displaystyle{ \frac{z \, dz}{(z-a)(z-b)} = \frac{dx}{x}. }[/math]
If [math]\displaystyle{ a\neq b }[/math], the solution is
- [math]\displaystyle{ x \frac{(z-a)^{a/(a-b)}}{(z-b)^{b/(a-b)}} = k }[/math]
where [math]\displaystyle{ k }[/math] is an arbitrary constant. If [math]\displaystyle{ a=b }[/math], ([math]\displaystyle{ (2A+B)^2 - 16C=0 }[/math]) then the solution is
- [math]\displaystyle{ x(z-a) \exp \left[\frac a {a-z}\right]=k. }[/math]
When one of the roots is zero, the equation reduces to Clairaut's equation and a parabolic solution is obtained in this case, [math]\displaystyle{ A^2+ AB -4C=0 }[/math] and the solution is
- [math]\displaystyle{ x(z\pm B)=k, \quad \Rightarrow \quad 4By = - AB x^2 - (k\pm Bx)^2. }[/math]
The above family of parabolas are enveloped by the parabola [math]\displaystyle{ 4By=-ABx^2 }[/math], therefore this enveloping parabola is a singular solution.
References
- ↑ 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.
- ↑ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
- ↑ Ince, E. L. (1939). Ordinary Differential Equations, London (1927). Google Scholar.
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