Similarly if u is a function of y only logu = (138) P and the requisite condition that the second member should be integrable is that L should be a function of y only. Ex. 1. (22 + y2 +2x) dx+2y dy = 0. (dp) - () Here " = 1; i logu = x. Ex. 2. (xy2 ex) dx — 22 y dy = 0. 394.] Again operating on the first two members of (134) in the manner indicated by the following results, we have du y dx - x dy (139) which is a solution of the differential equation; or in an apparently more general form, (%) = c. (142) Equation (141) is true also because simultaneously Pdx +q dy = 0, and PX+Qy = 0. If Qy-Px = 0, then from (140) we have also y dx + x dy = 0; is my = 6, (143) which is the solution of the differential equation; and in an apparently more general form, we have f (xy) = c. (144) Equation (143) is also true because we have simultaneously Qy-PX = 0, and P dx + Qdy = 0. 395.] If however neither PX+Qy nor Qy-P& vanishes, then from (139), we have du y dx — 2 dy s (dp) |dq) . (145) (145) ū = Pæ+ Qx Ilay) – deti now the right-hand member of this equation is an exact differential if P and Q are homogeneous functions of x and y of n dimensions. For in that case by Euler's Theorem, we have let+ y () = ne, * (e) + y (eky) = nq; and consequently as pdx +Qdy = 0, v { lehe) de + leden) dy} + { edx + (any) ay}= 0; and adding this to the numerator of the right-hand member of (145), and observing that de = (as) de + (y) dy, and dę = (dx + (.) dy, we have 2 dp+yda PX+Qy? (146) Pdx +qdy ; and therefore is an exact differential; and we have PX+QY pdx + Qdy = DU; (147) PX+Qy hereby then can u be determined, and an integral be found. Let these results be compared with Articles 379 and 380. If both members of (147) are multiplied by F(u), then we have [Pdx +Qdy Wol P(u) = /(u) du =)? 09 F'(u); (148) JPX +Qy and therefore if one integral of (147) can be found, an infinite number may also be determined. 396.] For another illustration of the theory of integrating factors let us take the first linear differential equation which is given in (64), Art. 382, and a means of integrating which has already been therein discussed. The form is {yf (2)— F(x)} dx + dy = 0. (152) On comparing this with Art 393, it appears that the condition requisite for the existence of a factor which is a function of x only is satisfied, so that we might deduce the form of the factor from that article. Let us however investigate it from first principles. Let u be the factor; therefore |