CHAPTER VIII. ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 1. IN the largest sense which has been given to the term, a singular solution of a differential equation is a relation between the variables which reduces the two members of the` equation to an identity, but which is not included in the complete primitive. In this sense, the relation obtained by equating to 0 some common algebraic factor of the terms of the equation might claim to be called a singular solution. dy d'y But, in a juster and more restricted sense, a singular solution of a differential equation is a relation between x and y, which satisfies the differential equation by means of the values which it gives to the differential coefficients &c., but is not included in the complete primitive. In this sense the equation x2+ y2 = n2, is a singular solution of the differential equation of the first order dy dx' d2 It reduces the members of that equation to an identity, but not by causing any algebraic factor of them both to vanish. At the same time it is not included in the complete primitive y − cx = n √√(1 +c2). And this is the juster definition, because that which is essential in the singular solution is thus in a direct manner connected with that which is essential in the differential equation. Def. Chap. I. When it is said that a singular solution of a differential equation is not included in the complete primitive, it is meant that it is not deducible from that primitive by giving to the arbitrary constant c a particular constant value. But although a singular solution is not included in the complete primitive, it is still implied by it. Upon the possibility of satisfying a differential equation by an infinite number of particular equations, each formed by the particular determination of an arbitrary constant, rests the possibility of satisfying it by another equation, to the formation of which each particular solution has contributed an element. We have seen in Chap. VII. how a singular solution, as representing the envelope of the loci defined by the series of particular solutions, possesses a differential element common with each of them. We shall now see that this property is not accidental —that it is intimately connected with the definition of a singular solution. It is important that the two marks, positive and negative, by the union of which a singular solution of a differential equation of the first order is characterized, and by the expression of which its definition is formed, should be clearly apprehended. 1st. It must give the same value of dy dx in terms of x and y, as the differential equation itself does. This is its positive mark, a mark which it possesses in common with the complete primitive, and with each included particular primitive. 2ndly. It must not be included in the complete primitive. This is its negative mark. Upon the analytical expression of these characters the entire theory of this class of solutions depends. Among the different objects to which that theory has reference, the two following are the most important. 1st. The derivation of the singular solution from the complete primitive. 2ndly. The deduction of the singular solution from the differential equation without the previous knowledge of the complete primitive. The theory of the latter process is so dependent upon that of the former that it is necessary to consider them in the order above stated. [Important additions to the present Chapter are given in the Supplementary Volume, Chapter XXI.] Derivation of the singular solution from the complete primitive. 2. The complete primitive of a differential equation of the first order, whatever may be the degree of the equation, is of the form $ (x, y, c) = 0. If we give to c a particular constant value in this equation we obtain a particular primitive. If we give to c a variable value by making it a function of x, or of y, or of both, we, as will immediately be shewn, convert the equation into any desired relation between x and y. We propose then to determine c as variable, but as so varying that the resulting relation between x and y shall continue to satisfy the differential equation. The general effect of the conversion of c into a function of x or of y must first be considered. may, by the conversion of c into a function of x, be transformed any desired equation containing x and y together, or y alone, but not into an equation involving x without y. into Let the desired result of transformation be (x, y) =0, or x (y) = 0, involving y at least. Combining either of these equations with the primitive we can eliminate y, and so obtain a relation between a and c which will determine c as the function of x required. It is evident however that the conversion of c into a function of a could not convert the primitive into an equation not involving y. For a variable cannot be eliminated from an equation, except by the aid of another equation which contains that variable. Similarly the conversion of c into a function of y would enable us to convert the given primitive into any desired equation involving, of the two variables, at least x. Ex. Let it be required to convert the equation y = cx into x2+ y2 = 1, by the conversion of c into a function of x. Eliminating y from the given and the proposed equation, we have This value of c substituted in y = cx, converts it into y = √ (1 − x2), which is equivalent to a2 + y2 = 1. 3. Let us now enquire what determination of c as a function of x will convert the primitive (x, y, c) = 0 into a relation between x and y still satisfying the differential equa tion. 1= Now the complete primitive of a differential equation of the first order is always by solution with respect to y reducible either to a single equation or to a series of equations of the form If we differentiate, regarding c as constant, we have as the derived equation dy df (x, c) .(2), and the elimination of c from this by means of the previous dy which satisfies the differential equation gives us a value of dx equation. That differential equation would then still be satisfied if c were regarded as variable, provided that the variation were such as to leave unchanged the form of the relation be |