nature of the curve remains the same, whatever are the values of X, and yo, yet the position of it alters; and consequently the differential equation expresses a property common to a series of curves, the particular one of which is determined by means of the arbitrary values X, and yo. But as a complete integral equation determines both the nature and position of the curve which it represents, it is plain that the coordinates of the first point must enter into the integral equation; and therefore the integral of (5) must contain these, and cannot be complete without them; the integral therefore of (5) must be definite; but it is convenient to leave the superior limits in the general form x,y, so that they may refer to any point on the curve. It is plain also, from the theory of definite integration, that if F(x, y) is the indefinite in. tegral of (5), the definite integral is F(x,y)-F (xo, Yo) = 0; (6) and as Xo, y, are arbitrary constants, we may replace F(x, y) by an arbitrary constant c; and thus the integral equation of (5) is of the form F(x, y) = C; (7) or, more generally, F(x,y,c) = 0; that is, in the process of integration one arbitrary constant c has been introduced. Again, suppose the given differential equation to be of the second order and of the form $(x, y, de, d*%) = 0; and let the inferior limits of integration correspond to the point (X., Y.), from which we will suppose the curve to begin: this point is of course arbitrary. Also since involves three consecutive points, see Art. 243, Vol. I, and as there is only one relation, viz. (8), between the three points, the second as also the first is arbitrary; but not so the third ; its position with reference to the other two becomes fixed by means of equation (8); and similarly will every other consecutive point on the curve, and thus the whole curve, become fixed; in the complete integral therefore the coordinates to these first two points must enter; and, by the theory of definite integration, in the form of two constants, which will be arbitrary, because the first two points on the curve are arbitrary: and thus if C1, C, are two arbitrary constants, the complete integral of (8) is of the form (9) P(x, y, C1, C2) = 0. This is also otherwise manifest : the first indefinite integral of (8) will be a function of x, y, and ; and therefore the definite integral will be of the form And replacing the second term by an arbitrary constant Cz, the first integral will be of the form and the integral of this will, by reason of what has already been said, involve another new arbitrary constant cq. It appears therefore that the complete integral of a differential equation of the second order requires the introduction of two arbitrary constants. By a similar process we may shew that n arbitrary constants enter into the complete integral of a differential equation of the nth order. It may perhaps be superfluous to remark that in thus taking the definite integral of a differential equation, the differentials or derived functions must not become infinite or discontinuous for any value of the variables between the limits. 366.] And to form a correct notion as to the meaning of a partial differential equation, let us consider the following example of a partial differential equation of the first order : (2-a) () +(y=0)( ) +(2-0)() = 0; or, as it may be otherwise and equivalently expressed, (–a)(x) + (7-6)(3) = z-c. (11) Equation (10) is the general equation of a tangent plane of a surface, which passes through a given point (a,b,c); or, what is equivalent, (10) implies that all the normals to the surface are perpendicular to straight lines which pass through a given point: and it is not for one surface only, or for one particular species of surface, that this property is true; it is not only for a given cone or for circular cones that the property holds good; but it is true of all conical surfaces of which the given point is the vertex: and (10) thus a symbol expressing a condition equally general must enter into the final integral equation : in other words, the complete in. tegral must contain the law of the director-curve of the conical surface; and such can be the case only when an arbitrary function is introduced: the complete integral therefore of a partial differential equation of the form (10) or (11) must contain an arbitrary functional symbol : in fact we know that the integral of (10) or (11) is either (12) (13) Hence it appears that the integral of a partial differential equation of the first order requires the introduction of one arbitrary function. Thus it appears that these geometrical explanations as to the introduction of constants and arbitrary functions in the case of total and partial differential expressions are in accordance with the reverse analytical process of Section 7, Chap. III, Vol. I. 367.] We may also thus prove that the complete integral of a differential equation of the nth order and first degree involves n arbitrary constants. Let us suppose the differential equation to be of the form p(x, y, .. = 0, and to admit of being put into the form day dy dn-ly). I. dx . dxn-1) (14) and let us suppose that it, and all its integrals up to the last, satisfy the conditions which are requisite for development in Taylor's series. Let (14) be differentiated successively, and let the necessary eliminations be performed, so that we can determine cont", dn+24 dyda-17 , ... in terms of x, y,; ; and let the limits of the integral of (14) be to, Yo and X, Y ; then, by equation (84), Art. 74, Vol. I, dayı (x - X.) + 1 day) (x - x)" y = yo+ 2 +...+1 +...; (15) \dx2 10 da"). 123 1.2 idux x - xo tl dxlo 1 where the subscript cyphers indicate particular values of the symbols; those, namely, which correspond to the inferior limit. • Now from the preceding remarks it is plain that all the differen id"-ly) tial coefficients after on-1), may be expressed in terms of to, yo, (am).... ( -), so that the series (15) will involve n and only n undetermined quantities, viz. the term independent of X, and the several coefficients of X, a, ... 2"-, which are n in number and may be expressed by n constants, C1, C2, ... Com; and therefore into the complete integral of (14) n arbitrary constants enter. Of course it is supposed that none of the quantities You infinite or discontinuous between the limits; as however no criteria are given for determining whether these conditions are satisfied or not, the above must be taken only to establish an a priori probability that the theorem, as stated, is true. A rigorous proof of a particular case will be given hereafter, and might be extended generally. · As an example of this process, let us take the equation which involves only one arbitrary constant, viz. Yo. 368.] When the integral of a given differential equation contains n arbitrary constants, and these in their most general form, it is called the general integral; and conversely, if an equation in terms of x and y satisfies a given differential equation of the nth order and contains n arbitrary constants, it is the general integral. If particular values are given to one or more of these arbitrary constants, as, for instance, if any of them is zero, then the integral is called a particular integral. Also sometimes one or more of the arbitrary constants may be replaced by a particular function of x and y, and the equation will still satisfy the given differential equation, when at the same time such a result cannot be obtained by giving any particular constant value to one or more of the arbitrary constants of the general integral : in this case the integral is called a singular solution. Our capital problem is the discovery of the general integral, by means of which particular integrals evidently may be determined. But we shall also investigate as far as possible the general properties of singular solutions, and indicate some specific forms of differential equations which admit of such solutions. In most cases we shall be obliged to leave the arbitrary constants undetermined; the complete integral of a differential equation requires that the integral should be definite, and therefore the constants ought to be expressed in terms of the limits; but it is manifest that this can be done only when the conditions of the problem are given, as in the geometrical applications of the calculus. Differential equations, however, for the most part arise in mechanics and other applied mathematics, on the investigation of which we have not yet entered: the constants therefore which are introduced in the process of integration must in most cases be left arbitrary, at least for the present. 369.] A simple form of differential equation which admits of integration immediately, or, as it is commonly said, by simple quadrature, is that where the variables are separated; in which case the expression contains the algebraical sum of several elements, each of which is a function of a single variable. The general form in the case of two variables is f(x) dx + $(y)dy = 0; (16) whence we have for the definite integral, x, and y, being corresponding values of x and y, (17) |