and the requisite condition that the second member should be

Ex. 2. (xy2- 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

(141)

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 ap

parently 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

αμ

du_ydz-zdy { (dr) - (de)};

μ

Px+QY dy

(145)

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

Р

{ (dx) dx + (da) dy } + y { (11) dx + (da) dy} = 0;

dy

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 r′(u), then we have

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

eff (x) dx {yƒ (x) − F(x)} dx + eff (x) dx dy

(153)

(154)

(155)

= c;

(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 c1 = '(c), where ' denotes an arbitrary function, from (155) we have

= eff(x)dx 6' { yes!(x)dx — ƒeƒƒ(4)dx y (x) da }

dx; (158)

And applying this most general value of μ, we have as the general integral of (152)

u = • {ye//(w)dx — fel/(xrde y (x) dx} = c. (159) — feff(x)dx

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. Ex. 1. dy+y dx = ax" dx.

[f(x)dx =

= x;

(161)

... (—)" n (n − 1)...8.2.1} e* +c;

.. y = a{x”—nx"− 1 + . . . ( — ) " n (n−1)...3.2.1}+ce ̄*, where c is an arbitrary constant.