and if the integrals are indefinite,

[f(x) dx + f$ (y) dy = c,

where c is an arbitrary constant.

(18)

And if there are three variables, the general form of the equaf(x) dx + (y) dy +x (z) dz = 0;

tion is

(19)

(20)

..sin+sin-1 y = sin-1c,

c being an arbitrary constant;

.`. x(1—y2)3+y(1−x2)3 = c,

which is the general integral; and if c = 0, we have a particular integral y = x.

370.] Another form in which the variables immediately admit of separation is

XY1 dx+YX1 dy = 0;

(21)

where x and x, are functions of x only, and Y and Y1 are functions of y only; for dividing through by X, Y1, we have

Ex. 3. If sin x cos y dx - sin y cos x dy = 0, then cosx = mcosy. Ex. 4. If (sec x)2 tan y dx + (sec y)2 tan x dy

=

0,

These methods however are so simple that it is unnecessary to add other examples.

SECTION 2.-Integration of exact total differentials of two and more variables.

371.] Let us first take the case of two variables, and suppose the differential expression to be

where P and Q are functions of x and y: it may be that (24) is the exact differential of some integral function of the form u = F(x, y) = c; or it may be that some factor common to the two terms has been divided out, and that (24) will not be an exact differential until this factor, or some other factor, has been introduced; this latter case is reserved to Section 6 of the present Chapter.

If (24) is the exact differential of the function

and as dx and dy are arbitrary, though infinitesimal, increments of x and y, (26) can only be true when

Hence we have a criterion whether (24) is an exact differential

d2u

d2u

or not; for since (dd) = (dody), if (27) are true,

dydx

and consequently if it is not on inspection plain whether (24) is an exact differential or not, we may apply the condition (28); and, if it is fulfilled, we are assured that (24) is an exact differential.

The equation (28) is commonly called the condition of integrability. Let us suppose it to be fulfilled. Since P dr is the

x-partial differential of u, the x-integral of P da will give the function of which enters into the general integral; and similarly the y-integral of qdy will give the function of y: the addition therefore to the x-integral of Pde of those functions of y which the y-integral of Qdy contains and which are not in the a-integral of P de will give the whole variable part of the general integral of (24); and the addition of a constant, or the determination of the definite integral, when the limits are given, at last gives the general integral of the differential equation.

Р

where y and x are functions severally of y and x only, and which are added to the partial integrals of Pdx and Qdy; and where Y is the sum of all the y-functions which are in (30) and are not in (29); and where x is the sum of the x-functions which are in (29) and are not in (30).

It will be observed that, if the variables are separated, as in (22), P and Q are functions severally of x and y only, and that

so that the criterion of integrability is satisfied; and thus the general integral is determined by two single integrations.

Also if P dx+Qdy can be divided into two parts, one of which is evidently an exact differential, it is sufficient to ascertain whether the other part is also such.

The following are examples of these processes;

so that the criterion of integrability is satisfied. Let u, and u, denote the x- and y-partial integrals; then

and let y and x be the functions of y and x respectively, which are added to the partial integrals as above; then

Y = cy2+ey; x = ax2+gx;

by means of either of which we have from (31), u = ax2 + bxy+cy2+gx+ey+k;

where k is an arbitrary constant.

Ex. 3.

{p(xy)+xy$′(xy)} dx + x2 p′(xy) dy = 0;

P = p(xy)+xy('(xy); .'. = 2xp′(xy)+x2y p′′(xy) ;

P=

de

dy

dq

= 2x p′(xy)+x2y¢"(xy);

dx

Q = x2 + (xy);

and thus the criterion of integrability is satisfied; and we have

372.] If (29) has been found in a definite form, the unknown function y in it may be determined without the integral (30). Take the y-differential of the definite integral of (29); viz.,

representing by Q, the value of Q, when x = x。:

Or representing P and Q by f(x, y) and 4 (x, y) we have

(33)

(34)

u =

=S2
[* f (x, y) dx + [ "$ (x, y) dy ;

yo

(35)

(36)

373.] Next let us take the case of a differential equation of three variables, and of the form

(37)

Pdx+Qdy + R dz = 0; where P, Q and R are functions of x, y and z. Now of course it may be that either (37) is an exact differential; or that some factor common to all the terms has been divided out, and that the expression can be made exact only by introducing this, or some other equivalent, factor: this latter case we shall, as heretofore, reserve to Section 6 of the present Chapter, and shall first consider the case where (37) is an exact differential. If we recognise immediately the general integral of (37), it is of the form, u = F(x, y, z) = c; and we need not apply criteria of integrability: and this is manifestly the case in such an example as,

whence

y2

c2

+ + — k2 = u = 0 ;
b2 c2

where k is an arbitrary constant; and in

(38)

If however the integral is not discoverable at first sight, still let us suppose (37) to be the exact differential of a function of three independent variables of the form (38); then