CHAPTER II.

ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES.

1. THE differential equations of which we shall treat in this Chapter may be represented under the general form

M and N being functions of the variables x and y.

In this mode of representation x is regarded as the independent variable and y as the dependent variable.

We may, however, regard y as the independent and x as the dependent variable, on which supposition the form of the typical equation will be

dx
M +N=0.
dy

For as any primitive equation between x and y enables us theoretically to determine either y as a function of x, or x as a function of y, it is indifferent which of the two variables we suppose independent.

It is usual to treat this equation under the form

Mdx+Ndy=0,

not however from any preference for the theory of infinitesimals, but for the sake of symmetry.

The order of this Chapter will be the following. As the solution of the equation, if such exist, must be in the form of a relation connecting x and y, I shall first establish a preliminary proposition expressing the condition of mutual depend

ence of functions of two variables; I shall then inquire what kind of relation between x and y is necessarily implied by the existence of a differential equation of the form

M+Ndy = 0;
dx

I shall discuss certain cases in which the equation admits readily of finite solution; and I shall lastly deduce its general solution in a series.

PROP. 1. Let V and v be explicit functions of the two variables x and y. Then, if V be expressible as a function of v, the condition

d V dv dV dv

dx dy dy dx

=

0

(1)

will be identically satisfied. Conversely, if this condition be identically satisfied, V will be expressible as a function of v.

1st. For suppose V (v). Then

=

And this is satisfied identically; since by the process of elimination the second member vanishes independently both of the form of v as a function of x and y, and of the form of Vas a function of v.

2ndly. Also if the above condition be satisfied identically, V will be expressible as a function of v. For whatever functions V and v may be of x and y, it will be possible by elimi

nating one of the variables x and

y

to express

Vas a function

of the other variable and of v. Suppose for instance the expression for V thus obtained to be

But, being by hypothesis an explicit function of both

identically. Therefore p (x, v), which represents V, does not contain a in its expression; and V reduces simply to a function of v.

We have supposed each of the functions V and v to contain both the variables x and y. But, whether this be or be not the case, the identical satisfying of (1) is the necessary and sufficient condition of the functional dependence of V

and v.

For suppose either V or v, and for distinction we shall choose v, to be a function of one of the variables only, as x, and V to be a function of v. Then is V also a function of x, dv dv

and as dy fied.

and vanish identically the condition (1) is satis

dy

Conversely, supposing v to be a function of a only, and (1) to be identically satisfied, that equation reduces to

of a primitive relation between x and y of the form

f(x, y) = c,

in which c is an arbitrary constant.

Let us first consider what is the immediate signification of the equation

M+Ndy

= 0.....

dx

We know that if Ax represent any finite increment of X, and

Ay the corresponding finite increment of

Ay
Ax

dy

y, will represent

dx

the limit to which the ratio approaches as Ax approaches

to 0.

Let us then first examine the interpretation of the equation

We have

M

Ay
Δα N

The second member of this equation being a function of x and y, since M and N are functions of those variables, we may write

the form of (x, y) being known when M and N are given.

Now if we assign to x any series of values, it is possible to assign a corresponding series of values of y, any one of which being fixed arbitrarily all the others will be determined by (3).

Thus let x, x1, x,... be the series of arbitrary values of x, and y, an arbitrary value of y corresponding to x, as the value of x, then, representing by Ax, the increment of x, i.e. the value which being added to x, converts it into x, we have by (3)

But as Ay, represents the increment of y, corresponding to
Ax, as the increment of x, it is evident that yo+Ay, will be
the value of y corresponding to x + Ax, as the value of x.
Representing then this value of y by y, we shall have

In like manner we shall find

Y2 = Y1+$ (x1, ÿ1) (x, − x ̧)..

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but, y, being already determined by (4), y, is determined, and continuing the operation, a series of values of y will be determined, only one of which is arbitrary, while all the others are assigned in terms of that arbitrary value and of the known values of x.

If, for example, we have the particular equation

Ay = (x + y) Ax,

3

-x

4'=x+y is linear,.. ex (√tedx + c )=x-1+Ce* and assign to x the series of values 0, 1, 2, 3, 4, &c., and at the same time assume that when x is equal to 0, y is equal to 1, we shall have the two following corresponding series of values, viz.

x=014=1 gives

&c.

By assigning a different value to y, or by assuming arbi-
trarily the value of some other term of the series yo, y1, Y2'
we should find another set of values of those quantities cor-
responding to the given values of x. But, in every such set,