Solutions of differential equations not obtained by giving particular values to the constant of integration in the general solution

Abstract

In considering the solution of Differential Equations, let the equation be taken in the form f(x,y,p)=c, in which p denotes dy/dx, and f is a rational, integral, and algebraic function of x, y, and p of degree n in p. It has been shown that, in general, this equation must have a solution in the form F(x,y,c)=0. F will always be a function of x, y, and a variable parameter, c. F will also be of degree n in c, but may not be, in all cases, a rational, integral, and algebraic function in x and y. We can assume f an indecomposable function. Then F will also be indecomposable. For if F could be factored, then to each of these factors would correspond a factor of f. There are, in some cases, solutions which can not be obtained by assigning particular values to the constant of integration in the general solution. Such a solution of a Differential Equation is called a Singular Solution.