The one-parameter group (15) will contain, in general, the identical transformation; i.e., for a certain value a. of a (15) will reduce to xi=x, yi=y. If now we denote by Ot an infini tesimal, and put in (15) a =---a.1-cdt. we shall find a transformation which transforms (x, y) into a point (xl, yi) such that the differences ..r.x=dx and y.--y=hy will be infinitesimals of the order of dt. This will be true unless cer tain exceptional cases arise which we need not, at present, discuss. From every one-parameter group we may deduce in this way an infinitesi mal transformation, and Lie has shown that conversely every infinitesimal transformation determines a one-parameter group. There is a similar connection between an r-parameter group and a corresponding set of r infinitesimal transformations, between which certain rela tions must then be satisfied.
A one-parameter group always has an in variant: i.e., there exists a function 0(x, y) such that, for all transformations (15) of the group. 0 (x,, yi)=E1(x, y). Such a function is said to admit the one-parameter group of trans formations. It admits, in particular, the in finitesimal transformation of the group. Simi larly, a differential equation may admit one or more infinitesimal transformations. Lie has shown that in the cases in which the variables may be separated, i.e., in which integration by quadratures is possible, it is possible to write down infinitesimal transformations which leave the equations invariant. He has developed a general theory showing what advantage is gained for the integration of a differential equa tion by the knowledge that it admits one or more infinitesimal transformations. Let us re mark, explicitly, that this theory is not con fined to equations of the first order nor even to ordinary differential equations.
Before passing to the consideration of the de mentary theory of equations of higher order, we proceed to explain the important notion of gular solution. Geometrically, an equation of the first order dy = f(x, y) determines the tan dx gent of an integral curve at every point of the plane. If we start from any point P, the tan gent of the integral curve passing through that point is completely determined. We follow the direction thus indicated for an infinitesimal dis tance to the point (x + ox, y +6y). At this point the tangent is again given by the differen tial equation, etc. We obtain in this way, syn thetically, the family of integral curves, say F(x, y, c) Any one of these curves is obtained by giving a definite value to the con stant of integration c. The envelope of this system of curves, however, will also be a solu tion of the differential equation. For if will also be a curve whose tangent satisfies the re quirements of the equation. But, in general, the envelope will not be itself a member of the family of curves, i.e., it will not be possible to find its equation by giving a special value to c. The envelope is then said to give a singular solution of the equation. If it exists, it may be found without any integration, that is to say, without a knowledge of the general integral of the differential equation.
The most important case of a differential equation of a higher order, which may be treated by elementary methods, is that of the linear homogeneous differential equation of the nth order with constant coefficients. A linear homo geneous differential equation of the nth order has the form dny (18) + + + dos If yi, y., , yn are particular solutions of the equation, y= co,-1- cnvn, where c,, . . . cn are constants, is also a solution. over, if y,, . . . yn are linearly independent, i.e., if they satisfy no relation of the form -1-ynyn= 0, where are constants, the above expression for y is the general solution. y,,. . . yn are then said to constitute a fundamental system of solutions. In the case that p,, . . . Pn are constants a fundamental system may be easily obtained. In fact we find that y=ePx is a solution of (18) if p is a root of the equation Pn + Pon' . + Moreover, if pl, . p,, are the roots, supposed distinct, of this equation, orhz, eftz actually form a fundamental system. If roots, say Pi, . pa, coincide, the a identical func tions el' , die are replaced by efrix, Total Differential In the case of an equation between two variables which we have considered so far, one important distinc tion, which we shall now have to make, has not been necessary. If P(x, y)dx Q (x,
is such an equation, it is always possible to find a single function ii(x, y) such that tis(x,Y) const. shall represent the general integral. Either the expression Pdx Qdy is the complete dif aa ferential of 40(x, y) so that P = ao and Q ax aY or else upon multiplication with Euler's in tegrating factor #(Pdx Qdy) becomes such a complete differential. This is not the case when there are more than two variables. Consider such an equation in three variables, (19) Pdx Qdy Rdz =0, where P, Q, R are functions of x, y, and 2. For the salce of symmetry assume that x, y., z are regarded as functions of a fourth variable t. The problem before us is to find all sets of functions x, y, z of t which will satisfy (19). It may happen that the left member of (19) becomes a complete differential upon multiplica tion with a function of x, y, 2, so that 430 = = 141? ax ay az The elimination of from these three equations shows that this can be the case only if P, Q, R satisfy the so-called integrability condition: , n(aQ aR , aR ap (2o) -r v az ay ax as +RCP ay ax Moreover it maY be shown that if P,Q,R satisfy this condition, there exists a function io(z-,y,z) and an integrating factor ,u(x, y, z) such that is(Pdx Qdy Rdz) = do, so that integration of (19) will give the result s6(x, y,z) const. But if (20) is not satisfied, no integration of (19) in this sense is possible. The reason for this distinction as well as the discussion of the non-integrable case will be clearly understood if we make use of a geo metric interpretation. Let x, y, be Cartesian co-ordinates of a point in space. If x, y, are lmown as functions of t, there will be determined a certain space-curve. It is our problem to de termine such space-curves x f (0, y=g(t), z h(t) as satisfy (19). Through every point (x., ye, zol of space there may be drawn an infinity of such curves. The tangents of all of these curves which pass through the point (xo, yo,zo) form a plane pencil with (xo, yo, 20) as vertex and the plane P(x., zo) (x xo) Q Yo. zo) (Y Yo) R (x., y., zo) (z z,) = 0 as plane. Thus there is for every point P a plane p containing P, to which all of the in tegral curves of (19) which pass through P must be tangent. We may now imagine an integral curve of (19) constructed as follows: Start from a given point P and construct the corresponding plane p. We go from P to a point Q infinitesimally close to P but otherwise arbi trarily situated in the plane p. At Q we con struct the plane q corresponding to it, and in this plane we pick out a point R infinitesimally close to O. Proceeding in this way we grad ually build up an integral curve. It may happen that oll of the integral curves of (19) which pass through the point P are situated upon a certain surface S. If this is the case for all points P, the integrability condition is satisfied; there exists a single infinity of sur faces 4(x, y, z) = c, such that an arbitrary curve upon each of these surfaces satisfies the differential equation. In general, however, such a family of surfaces does not exist. We may then integrate (19) as follows: Take an ortn trary surface 0(x, y, 2) =O. Let P be any point upon it. Let p be the plane of the pencil of direc tions which the differential equation assigns to P, and let p' be the plane tangent to the surface co(x, y, 2) =0 at P. The intersection t of p and p' will be at the same time tangent to an in tegral curve of (19) and tangent to the surface 1, =O. From P we go along t to a point Q in finitesimally close to P and there repeat this process. We may build up in this way all of the integral curves of (19) which are situated upon an arbitrary surface- Upon every arbi trary surface there will be a single infinity of such curves. Analytically this process may be carried out as follows: From 11)=0 we find aip , _,_ ax ay as= O. ax ay From this equation and IP 0, ds and s may be expressed in terms of x, y, dx and y. Substitu tion of these values into (19) gives rise to an equation of the form 111(x, y)dx+N(x, y)dy=----0, which may be integrated, in the form 0 (x, y) =c. This latter equation together with IP(x, y, 2)=0 gives the required solution. By giving all possible forms to the functions IP all possible solutions will be obtained.