4. The converse form of the property last noticed is of sufficient importance to be stated as a distinct proposition, namely,

Prop. II. If U and u be functions of x and y, and Udu be an exact differential, then U will be a function of u.

Hence the second member being an exact differential we have by Prop. I.

Therefore, by the proposition in the first Article of the second Chapter, U will be a function of u.

7. {n cos (nx + my) — m sin (mx + ny)} dx

+ {m cos (nx+my) — n sin (mx + ny)} dy = 0.

8. Shew, without applying the criterion, that the following are exact differentials, viz.

distinguishing between the different cases which present themselves according, 1st, as b and c are of the same or of opposite signs; 2ndly, as a is equal to, or not equal to, 0.

11. Shew by the criterion that the expression

is generally an exact differential, and exhibit the functional

dM dN

forms which and assume.

dy dx

CHAPTER IV.

ON THE INTEGRATING FACTORS OF THE DIFFERENTIAL

EQUATION Mdx+Ndy=0.

1. THE first member of the equation Mdx + Ndy = 0 not being necessarily an exact differential, analysts have sought to render it such by multiplying the equation by a properly determined factor.

Thus the first member of the equation

(1+ y2) dx + xy dy = 0

is not an exact differential, since it does not satisfy the condM dN

dition =

dy dx

but it becomes an exact differential if the

equation be multiplied by 2x, and its integration, which then becomes possible, leads to the primitive equation

x2 (1+ y2) = c.

The multiplier 2x is termed an integrating factor.

We propose in this Chapter to demonstrate that integrating factors of the equation Mdx+Ndy = 0 always exist, to investigate some of their properties and relations, and to shew how in certain cases integrating factors may be discovered. To complete this subject we shall, in the next following Chapter, investigate a partial differential equation, upon the solution of which their general determination depends, and shall examine some of the conditions under which the solution of that equation is possible.

2. To every differential equation of the form

Mdx+Ndy = 0,

pertains an infinite number of integrating factors, all of which are included under a single functional expression.

It has been shewn, Chap. II. Art. 2, that the above equation always involves the existence of a complete primitive of

dy

this equation must be the same as the value of furnished by

dx

Let be the value of each of these ratios, then

μ

As μM and μN are therefore the partial differential coefficients with respect to x and y of the same function (x, y), the expression Mdx+μNdy will be an exact differential. Thus Mdx+Ndy is always susceptible of being made an exact differential by a factor μ.

3. The form of the complete primitive is however without gain or loss of generality susceptible of variation. Thus the primitive x2 (1+ y) = c, Art. 1, might, without becoming more or less general, be presented in the forms

2

sin {x2(1+ y2)} = c,1, log {x2(1+ y2)} = C2,

or in the functional form fa*(1+ y)}=c, where c, c,, c, are arbitrary constants. And generally a complete primitive expressed in the form V-c may be expressed also in the form f(V) = c, f(V) denoting any function of V. These variations in the form of the complete primitive imply corresponding variations in the form of the integrating factor, a special determination of which has already been given, Art. 1.

To investigate the general form under which all such special determinations are included, let us suppose μ to be a particular integrating factor of Mdx+Ndy, and let μMdx+μNdy bé the exact differential of a function (x, y). Then representing for the present (x, y) by v, we have

μMdx+μNdy = dv.

Multiply this equation by ƒ(v), an arbitrary function of v; such being, by Art. 4, Chap. III., the general form of a factor which will render the second member an exact differential. We have

μf (v) (Mdx + Ndy) = ƒ (v) dv.

Now the second member of this equation being an exact differential the first is so also. As moreover the first member of the above equation can only become an exact differential simultaneously with the second, the factor f(v) is the general form of a factor which renders Mdx +Ndy an exact differential.

We may express the above result in the following theorem.

μ

If u be an integrating factor of the equation Mdx+Ndy=0, and if v = c be the complete primitive obtained by multiplying the equation by that factor and integrating, then uf (v) will be the typical form of all the integrating factors of the equation.

Furthermore, f(v) being an arbitrary function of v, the number of such factors is infinite.