CHAPTER XXV.

ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSOCIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS.

1. THE term simultaneous is here applied to a system of partial differential equations, to signify that in that system there is but one dependent variable, the general expression of which, as a function of the independent variables satisfying all the equations at once, is the object of search. All linear partial differential equations of the first order being reducible to the homogeneous form, we shall presuppose this reduction here. Under this form indeed the problem actually presents itself in Geometry, in the theory of partial differential equations of the second order, and in Theoretical Dynamics.

We are sometimes led, in connexion with the same class of inquiries, to systems of ordinary differential equations marked by the peculiarity that the number of the variables exceeds by more than one the number of the equations. Such systems are intimately connected with the former-stand to them indeed in a similar relation to that which the Lagrangean auxiliary system bears to the single partial differential equation from which it arises. The theory which explains this connexion, and grounds upon it the method of solution of both systems will form the subject of the present Chapter.

Connexion of the Systems.

2. PROP. I. The solution of a system of simultaneous linear partial differential equations of the first order may be

ART. 2.] LINEAR PARTIAL DIFFERENTIAL EQUATIONS.

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made to depend upon that of a system of ordinary differential equations of the first order in which the number of the variables exceeds by more than one the number of the equations.

The system of partial differential equations being reduced to the homogeneous form, Chap. XXIV. Art. 6, let n be the number of the equations, x,,,,..... the independent variables, and P the dependent variable.

Then from the n given equations determining

dP dP
dx' dx,'

dP

dxn'

we obtain an equivalent system of equations which, by transposition of its terms to one side, assumes the reduced form

a single partial differential equation which, on account of the arbitrariness of λ,, λ,, ...... λ, is equivalent to the system. from which it was formed.

λ,,,,

whence, eliminating A,, A,, ......, we have the system of ordinary differential equations

These equations being included in the previous system (3), any integrals

of them will be integrals of it. Therefore u, v, w,... will be values of P satisfying the partial differential equation (2). For they will be the only values which can satisfy it independently of A, A, ...... A. Hence they will satisfy the equivalent system (1), and the general integral of that system will be

F(u, v, w, ...) = 0

the form of F being arbitrary.

(5),

Thus the relation of the system (4) to the system (1) is the same as the relation of the auxiliary system of a single linear

The

partial differential equation to that equation. And the ground of this relation is seen to be the same in both cases. one form necessitates the other.

3.

Instead of employing the above mode of deducing the auxiliary system, we might employ the following which is practically more convenient.

Since any value P which satisfies the partial differential equations determines Pc as an integral of the ordinary system, the latter must be consistent with dP=0 in its developed form

by means of the n given equations (1), we have

In the same way we can pass from the system of ordinary to that of partial differential equations. From the equation dP=0, in its developed form, we must eliminate a number

of differentials dx,, dx,,... equal to that of the given equations, and then equate to 0 the coefficients of the remaining differentials.

4. Lastly, the formal connexion of the two systems should be noticed. The partial differential equations being given in the reduced form (1), the ordinary system may be constructed as follows: For any differential coefficient, as in any

dP

" dx n+1 column after the first, write the corresponding differential dxn+1, subtract from this the sum of dx1, dx, ...... din dx, multiplied respectively by the descending coefficients of that column, and equate the result to 0. The system of equations thus successively formed will be the auxiliary system sought.

The transition from the ordinary to the partial system may be effected by the same rule, substituting only differentials for differential coefficients.

[It appears from the manuscript that an example was to have been supplied here.]

Up to this point the theory of systems of partial differential equations is in analogy with that of single equations. But here a difference arises. We do not know beforehand what number of integrals a system of ordinary differential equations, in which the number of variables exceeds by more than one the number of the equations, admits.

The theory which removes this difficulty will be developed in the following sections. It will be shewn that a system of linear partial differential equations which admits of solution by the assigning to the dependent variable a value which satisfies all the equations in common, must either itself satisfy a certain condition, or be capable of being developed into a new but equivalent system which will satisfy that condition. It will be shewn that when that condition is satisfied, the auxiliary system of ordinary, is capable of expression as a system of exact differential equations determining the integrals sought.