Isabel Maddison

On Singular Solutions of Differential Equations of the First Order and the Geometrical Properties of Certain Invariants and Covariants of their Complete Primitives Quarterly Journal of Pure and Applied Mathematics, Vol. 28 (1896), 311-374

Introduction

The Theory of Singular Solutions is now generally understood to relate to certain solutions of a rational, integral, algebraic equation of the nth degree in p, (dy/dx), whose coefficients are rational, integral, algebraic functions of x, y. The primitive of this equation is known to be a rational, integral, algebraic equation of the nth degree in an arbitrary constant, Ω, having coefficients which are functions, not necessarily algebraic, of x, y. A solution of the p-equation which cannot be deduced from the primitive by giving Ω any special value is called a "singular solution"; a solution which can be so deduced is called a "particular integral." (We shall denote this by the letters P.I.).

The singular solution is known to be the envelope of the family of curves represented by the complete primitive; hence the geometrical statement of the above is: A rational, integral, algebraic equation of the nth degree in p, whose coefficients are rational, integral, algebraic functions of x, y, represents a family of curves, not necessarily algebraic, depending on the arbitrary parameter Ω, and such that through every point of the plane there pass n curves of the family. The equation of an envelope of this family of curves satisfies the p-equation and is called a "singular solution"; any particular curve of the family is a "particular integral."

If the coordinates of any particular point be substituted for x, y in the Ω- and p-equations these equations give respectively, the n values of Ω which determine the n curves through the point, and the n values of p which determine the directions of the tangents to the n curves at the point. If the particular point be on the envelope, two of the curves through the point are consecutive, and the directions of the tangents to these two curves are consecutive, hence each of the equations has a pair of equal roots, that is, the envelope is the locus of points at which the Ω-equation has a pair of equal (consecutive) roots in Ω, and the p-equation has a pair of equal (consecutive) roots in p.

But regarding the equations as binary quantics in Ω/1 and p/1 respectively, the discriminant of the Ω-equation is the locus of points at which two values of Ω are equal, and the discriminant of the p-equation is the locus of points at which two values of p are equal; therefore we get, as a general result which will be modified later, a common factor of the Ω- and p-discriminants is a singular solution of the p-equation and gives the envelope of the family of curves represented by it.