THEORY OF EQUATION S ; G ALOI S' T H EORY ) .
Moreover, even if the reduction to quadratures can be effected, such a reduction is, properly spealdng, the beginning and not the end of the investigation. For it does not suffice to give a formal indication of the relation between x and y; this relation must be thoroughly under stood in its essential properties before the in tegration can be said to have been accomplished. Nevertheless the consideration of the simpler cases, in which integration by means of elemen tary functions or by quadratures is possible, constitutes a first important chapter of the theory of differential equations. We may char acterize this chapter as the elementary theory of differential equations.
Elementary Theory of Differential Equa tions.--We have already referred to the case in which the variables are separated. In many cases a simple transformation will accomplish the separation. Consider, for example, the equation dy (4) P- - F y = 0 , dx where P is a function of x only. We may write dy Pdx=0, whence log y Pdx=log c, or (5) y=ce P f dX, This example will be useful in enabling us to treat, at once, a more general equation ; we shall do so, moreover, by maldng use of a method frequently employed, and especially important in the applications to theoretical astronomy, the method of variation of constants. We consider the equation dy (6) dx where P and Q are functions of x only. This equation is the most general linear differential equation of the first order, a linear equation being one which contains y and its derivatives in no higher than the first power. Equation (6) differs from (4) only in having Q in the right member in place of zero. The expres sion (5) will certainly not satisfy (6) since it satisfies (4). Clearly, however, it must be possible to satisfy (6) by an expression of the form analogous to (5), viz., (7) y=ue where u is a properly chosen function of x instead of being a constant. Moreover, as we shall see, we can actually determine the func tion u by quadratures. In fact, we find from (7) dy (du u e dx dx which gives, on substitution into (6) du _Qef dx so that we shall have Pdx (5) y=e [C f Qe" dx1' as the general integral of (6). This formula was found by Jacob Bernoulli, who also showed that the equation dy vs+ (9) d P Y =QY could be reduced to (6) by putting u =yin. The homogeneous equations of the form (10) dy I L\ dx kx/ where ss () depends only upon the ratio of y to x, may be solved by quadratures. In fact, if we put 3-vx, the equation becomes dx dv x vG(v) whence dv (11) log x iv yo(v) Eulees method of the integrating factor is sometimes useful. It rests upon the following considerations. Let se (x, onst. be the
equation of any integral curve of the equation (12) P(x, y)dx +Q(x, y)dy-=0.
We shall have, by differentiation from y)=const., ait, a0 -r uy-=. 0, ax ay an equation which must have the same signifi cance as (12). We must, therefore, have 84) (13) pis pQ(x, , (x, y)= aX aYif Ai is a properly chosen function of x and y. If Is is known, the determination of se bv quad ratures can be immediately accomplished on account of the two equations (13). For this reason IA is called an integrating factor. Equa tions (13) show that Ai must satisfy the partial differential equation a(liP) aogQ) ay ax In general, the deterrnination of an integrating factor is just as difficult as the integration of the equation. But Euler succeeded in finding a number of equations with known integrating factors. Herein lies the value of the method.
By means of these various methods there was obtained. in the course of time a consider able number of equations whith could be inte grated by quadratures. Lie showed that this rather scrappy theory could be understood as the consequence of a single principle. This we shall now proceed to explain, making use of geometric images for the salce of clearness as well as brevity.
The equations y), Y), are said to constitute a transformation of the point (x, y) into the point (xi, ya if they can be solved for xi and These equations may contain a certain num ber of arbitrary constants ch, . . . ar; they are then said to constitute an r-parameter family of transformations. Let us consider the simplest case of a one-parameter family which we may write (15) = 56(x, y; a), y2=IP(x, y; a). If the parameter a has a definite value, this transformation converts every point (x, y) into a definite other point (x,, y,). Let us transform this new point (x,, y2) by equations of the some form, but with a different parameter b, into a third point (x., y.), so that we shall have (16) x2= 9(x2, y,; b), y2=1P(xl, y,; b). In general, if we eliminate xi, y, between (15) and (16) we shall find .r, and y, as functions of .r, y, a and b. It may happen that these func tions assume the form (17) x2=0(x, y; c), y2=IP(x, y; c) where c is a function of a and b, and where the functions it and IP are the same as in (15) and (16). If this is the case, the transformations. (15) are said to form a one-parameter group. The one-parameter family of transformations (15) then has the property that the transforma tion, obtained by combining any two of its transformations, is itself a member of the family. It is for this reason that the family is then called a group. (See GROUPS, THEORY OF). It is obvious how this definition may be ex tended to cover r-parameter groups.