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INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO OR MORE
Functions of the first order.
In order that a differential function of two variables of the first order, such as
Pdx + Qdy, should be the differential of a function u, it is necessary that the condition
dy - dx should exist. When this criterion of integrability holds good, we find
u = [Pdx + sdy (Q - SPdx);
The application of these formulæ may be generally facilitated by observing that in the second term of the former it is only necessary to integrate the terms in Q which involve x only, and in the latter those terms of P which involve y only.