and the requisite condition that the second member should be Ex. 2. (xy-ex3) dx-x2y dy = 0. 394.] Again operating on the first two members of (134) in the manner indicated by the following results, we have If Px+Qy = 0, then from (139) we have also which is a solution of the differential equation; or in an apparently more general form, Pdx+Q dy = 0, and P+Qy = 0. If qy-Px = 0, then from (140) we have also (142) (143) which is the solution of the differential equation; and in an apparently more general form, we have Equation (143) is also true because we have simultaneously QY-Px = O, and Pdx+Qdy = 0. 395.] If however neither P+Qy nor Qy-Pa vanishes, then from (139), we have 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 = 2 {(d) dz + (de) dy} + y { (d) dx + (de) dy} = 0; dx dy and adding this to the numerator of the right-hand member of (145), and observing that 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 r(u) = f¥'(u) du = f3dx+Qy and therefore if one integral of (147) can be found, an infinite number may also be determined. Hence, if P and Q are homogeneous functions of n dimensions, (Px+Qy) is an integrating factor of Pdx+qdy = 0. The following are examples of integration by this process. Ex. 1. 2xy dx + (y2x2) dy = 0. Here P+Qy = x2y + y2; 2 xy dx + (y2 — x2) dy = DU 0; x2 + y2 = log ; y = c, y where c is an arbitrary constant. Therefore by reason of (148) the most general integral is the right-hand member of which is an exact differential if P = yf(xy), Q = xp (xy); for in this case And thus if P = yf(xy), Q = x(xy), (Px—Qy)-1 is an inte 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 {y f(x) − 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. multiplying (152) by this value of μ we have eƒƒ (x) dx {y ƒ (x) — F(x)} dx + eff (x) dx dy = 0; - = [ eff(x)dx {yƒ (x)—r(x)} dx ; ,, = (153) (154) (155) (156) (157) where c is an arbitrary constant, and is the second arbitrary constant introduced in the integration of the series of equations given in (154); so that as c, = '(c), where ' denotes an arbitrary function, from (155) we have H = essundee { yessuar — [essinary(x)dx}{ dx (158) And applying this most general value of μ, we have as the general integral of (152) · { yes/(x) dx — [ ef/(x)dx p (x) dx } = c. (159) — It appears then that the equation (152) when multiplied through by eff(x)dx is an exact differential, and may be integrated as such; this is also otherwise evident; since dy+yf(x) dx = F(x)dx ; eff(x)dx dy+yeff (x)dx f (x) dx = eff(x)dx F(x)dx; (160) whence, as the left-hand member is an exact differential, by integration we have which result is the same as that of Art. 382. The following are examples of this process. [f(x)dx = = x; (161) .. y = a {x" — nx"-1 + ... (—)"n(n-1)...3.2.1}+ce*, where c is an arbitrary constant. |