(3). 125 Now as the original equation gives p = (x − 1)3, the complete primitive found by substitution of this value in (3) will be and it would be directly obtained in this form by integrating the original equation reduced by algebraic solution to the form This example illustrates the process but not its advantages. Ex. 2. Given x=1+p+ p3. Here dy =pdx = pdp + 3p3dp; 2 p2, 3p1 therefore y= + +c.. .(5), between which and the original equation p must be eliminated. We may do this so as to obtain the final equation between x and y in a rational form; but, if this object is not deemed important, we may, by the solution of a quadratic, determine p from (5) and substitute its value in the given equation. C' being an arbitrary constant introduced in the place of 1-3c; and y will be found by substituting this value of p in the original equation. Equations in which x and y are involved only in the first degree, the typical form being xp (p)+y+ (p) = x (p). 6. Any equation of the above class may be reduced to a linear differential equation between x and p, after the solution of which, p must be eliminated. The reduced equation is found by differentiating the given equation and then eliminating, if necessary, the variable y. It may happen that such elimination is unnecessary, y disappearing through differentiation. Ex. Let us apply this method to the equation. The second of these, which alone contains differentials of the new variables x and p, is the true differential equation between x and p. But what relation does the rejected equation (2) bear to the given differential equation (1), and what relation to its complete primitive just obtained? If we eliminate p between (1) and (2) we obtain a new relation between x and y not included in the complete primitive already found, i. e. not deducible from that primitive by assigning a particular value to its arbitrary constant, and yet satisfying the same differential equation, and, as we shall hereafter see, connected in a remarkable manner with the complete primitive. Such a relation between x and y is called a singular solution. We shall enter more fully into the theory of singular solutions in a distinct Chapter, but the following example will throw some light upon their nature, as well as illustrate the process above described. and this value substituted in the original equation gives, after freeing the result from radical signs, the singular solution. y2 = 4mx .(6), Here the singular solution (6) is the equation of a parabola whose parameter is 4m, and the complete primitive (5) is the well-known equation of that tangent to the same parabola which makes with the axis of x an angle whose trigonometrical tangent is c. dy Now, for the infinitesimal element in which the curve and its tangent coincide, the values of x, y, and are the same in both. And thus it is that the algebraic equations of the curve and of its tangent satisfy the same differential equation of the first order. On the other hand, if (5) be regarded as the general equation of a system of straight lines, each straight line in that system being determined by giving a special value to c in the equation, the envelop or boundary curve of the system will be determined by (6). Here the singular solution is presented as the equation of the envelop of the system of lines defined by the complete primitive. 7. In the second place let us consider the more general equation y=xf (p)+(p). a linear equation of the first order by which x may be determined as a function of p. The elimination of p between the resulting equation and the given one will give the complete primitive. The typical equation xp (p)+y¥ (p) = x (p) may be reduced to the above form by dividing by (p), but it may also be treated independently by direct differentiation. Ꮖ Instead however of forming a differential equation between x and p, we may form a differential equation between y and p. Or, with greater generality, representing any proposed function of p by t, we may form a differential equation between either of the primitive variables and t. Such a differential equation will necessarily be linear with respect to the primitive variable retained, and its solution must of course be followed by the elimination of t. And this general procedure, more fully to be exemplified when we come to treat of some of the inverse problems of Geometry and of Optics, is often attended with signal advantage. Ex. Given x+yp = ap2, We shall reduce this to a differential equation between x and p. then eliminating y by means of the given equation, we have 1+p2+ (ap2 — a) dr dp dp = which may be reduced to the linear form dx' V |