454
DIFFERENTIAL
to verify in examples that this is the case in at least the majority of the differential equations which are known to be integrable in finite terms. As space forbids 'tion of1' ^6 attempt, we give a theorem of very general theory of application for the case of a simultaneous continu- complete system of linear partial homogeneous toSforma^S (^^'erent^a^ equations of the first order, to the theories, solution of which the various differential equations discussed have been reduced. It will be enough to consider whether the given differential equations allow the infinitesimal transformations of the group.
EQUATIONS
similar result follows that it can he integrated by quadraturesBut if the group of three parameters be simple, this result must be> replaced by the statement that the integration is reducible to quadratures and that of a so-called iRiccati equation of the first order, of the form dy/dx=A + By + Cy , where A,B,C are functions of x. (4) Similarly for the integration by quadratures of an ordinary equation yn=^(xyy1 • • yn-i) of any order. Moreover, the group allowed by the equation may quite well consist of extended contact transformations. An important application is to the case where the differential equation is the resolvent equation defining the group of transformations or rationality group of another differential equation (see below); in particular, when the rationality group of an ordinary linear differential equation is integrable, the equation can be solved by quadratures. Following the practical and provisional division of theories of differential equations, to which we alluded at starting, into transformation theories and function theories, we pass now to give some account Cons/derof the latter. These are both a necessary logical ^UacUon complement of the former, and the only remain- theories of ing resource when the expedients of the former differenhave been exhausted. While in the former e^a^01,s investigations we have dealt only with values of the independent variables about which the functions are developable, the leading idea now becomes, as was long ago remarked by G. Green, the consideration of the neighbourhood of the values of the variables for which this developable character ceases. Beginning, as before,, with existence theorems applicable for ordinary values of the variables, we are to consider the cases of failure of such theorems. When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied. We have had examples in systems of linear equations, and in the case of a set of equations px = t/q, • •, pr = (f)r. For the case when the number of equations is the same as that of dependent variables, the following is a general theorem which should be referred to : Let there be r equations in r dependent variables z1 • • zr and n independent variables x1- • xn; let the differential coefficient of zff of highest order which ^xtetence enters be of order Zq., and suppose dh<rz(T/dxlh<r theorem. to enter, so that the equations can be written dhrzjdx-^o- = <Ba, where in the general differential coefficient of zp which enters in say dki+ ‘' +knzpf dxxki ■ • dxnkn, we have /t1< hp and k1 H 1- len<_hp. Let al • • arfx • • hr and be a set of values of xx • • xnzx • • zr and of the differential coefficients entering in about which all the functions dq ■ • dv are developable.. Corresponding to each dependent variable za, we take now a set of Zq. functions of aq • • xn, say cj)a, c£(r(1,, • • •, cf)Jh _1V arbitrary save that they must be developable about a2a5 • • an, and such that for these values of x2 - • xn, the function <j)p reduces to bp, and the differential coefficient
- 2+ • • +kncf)plki}/dx2k2 ■ • dxnkn reduces to bkf ■ •
Then the theorem is that there exists one, and only one, set of functions zx- • zr of xx ■ • xn developable about ax- • an satisfying the given differential equations, and such that for xx = ax we have z^ = dzaldxx — • • • dhr~1z(r/dh<r~1xx her = <t>J ~ L And, moreover, if the arbitrary functions• • • contain a certain number of arbitrary variables tx • • tm, and be developable about the values tx° • • tm° of these variables, the solutions zx- • zr will contain tx • • tm, and be developable about tx° • • tm°.
It can be shown easily that sufficient conditions in order that a complete system IIj/=0 • • 11*/= 0, in n independent variables, should allow the infinitesimal transformation I/= 0 are expressed by k equations IkP/-PII/=XilII1/+ • • +i!cnkf. Suppose now a complete system oi n-r equations in n variables to allow a group of r infinitesimal transformations (Pj/, • • , Pr/) which has an invariant subgroup of r - 1 parameters (l//, . • , Pr_j/), it being supposed that the n quantities IIj/, • • , IIn_r/, Pj/, . . ’ Pr/ are not connected by an identical linear equation (with coefficients even depending on the independent variables). Then it can be shown that one solution of the complete system is determinable by a quadrature. For each of lid//- P^H,./ is a linear function of IIj/, • • , IIM-r/ and the simultaneous system of independent equations IIj/ = 0 • • II„_r/= 0, Pj/= 0 ■ • Pr_j/=0 is therefore a complete system, allowing the infinitesimal transformation Pr/. This complete system of n — equations has therefore one common solution w, and Pr(w) is a function of w. By choosing w suitably, we can then make Pr(w) = l. From this equation and the n- equations 11^ = 0, P„w = 0, we can determine w by a quadrature only. Hence can be deduced a much more general result, that if the group of r parameters be integrable, the complete system can be entirely solved by quadratures ; it is oidy necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n-r equations in » — 1 variables, subject to an integrable group of r-1 parameters, and to continue this process. We give some examples of the application of the theorem. (1) If an equation of the first order y'=p(x, y) allow the infinitesimal transformation Wldx + ydf'jdy, the integral curves w{x, y) = y°, wherein u(x, y) is the solution of p(x, y) J^==0 reducing to y for x=x°, are interchanged among themselves by the infinitesimal transformation, or w {x, y) can be chosen to make £dujdx + yduijdy = 1 ; this, with dwldx-Vfdwdy = f), determines w as the integral of the complete differential {dy - pdx)l(y This result itself shows that every ordinary differential equation of the first order is subject to an infinite number of infinitesimal transformations. But every infinitesimal transformation ^df/dx + ydfldy can by change of variables (after integration) be brought to the form df/dy, and all differential equations of the first order allowing this group can then be reduced to the form F(x, ely/dx) = 0. (2) In the case of an ordinary equation of the second order y" = p{xyy'), equivalent to dyjdx=yl, dyildx=//(xyy1), if J H,H1 be the solutions for y and yx chosen to reduce to y° and y{ when x=x°, and the equations H=y, H1=?/1 be equivalent to o)=y°, = yf, then w, w1 are the principal solutions of H/= dfjdx + yidf/dy + 'pdf/dyl = 0. If the original equation allow an infinitesimal transformation whose first extended form (see art. Groups) is P/ = ^dffdx + ridf/dy + riff fly where yft is the increment. of dy/dx when £5£, yot are the increments of x, y, and is to be expressed in terms of x, y, yx, then each of Pw and Pwj must be functions of w and w,, or the partial differential equation II/ must allow the group P/. Thus by our general theorem, if the differential equation allow a group of two parameters (and such a group is always integrable), it can be solved by quadratures, our explanation sufficing, however, only provided the form II/ and the two infinitesimal transformations are not linearly connected. It can be shown, from the fact that r]x is a quadratic polynomial in yx, that no differential equation of the second order can allow more than 8 really independent infinitesimal transformations, and that every homogeneous linear differential equation of the second order allows just 8, being in fact reducible to d2y/cfo2 = 0. Since every group of more than two parameters has subgroups of two parameters, a differential equation of the second order allowing a group of more than two parameters can, as a rule, be solved by quadratures. By transforming the group we see that if a differential equation of the second order allows a single infinitesimal2 transformation, it can be transformed to the form M{x,dyldx, dxy/dx ) ; this is not the case for every differential equation of the second order. (3) For an ordinary differential equation of the third order, allowing an The proof of this theorem may be given by showing that if integrable group of three parameters whose infinitesimal trans- ordinary powrer series in xx - ax • • xn — an tx — tf • • tm - tm° be sub-0 formations are not linearly connected with the partial equation to stituted in the equations wherein in the coefficients of (xx - aj1 . which the solution of the given ordinary equation is reducible, the x1 •, (x1 - a/V”1 are the arbitrary functions </>„</>/• • • ~K
