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