The Integrating Factor

Let

be a differential equation which is not exact. The theoretical method of integrating such an equation is to find a function such that the expression

is a total differential . When has been found the problem reduces to a mere quadrature.

The main question which arises is as to whether or not integrating factors exist. It will be proved that on the assumption that the equation itself has one and only one solution,¹ which depends upon one arbitrary constant, there exists an infinity of integrating factors.

Let the general solution be written in the form

where is the arbitrary constant. Then, taking the differential,

or, as it may be written,

Since, therefore,

is the general solution of

the relation

must hold identically, whence it follows that a function exists such that

Consequently

that is to say an integrating factor exists.

Let be any function of , then the expression

is exact. If, therefore, is any integrating factor, giving rise to the solution , then is an integrating factor. Since is an arbitrary-function of , there exists an infinity of integrating factors.

Since the equation

is exact, the integrating factor satisfies the relation

or

Thus satisfies a partial differential equation of the first order. In general, therefore, the direct evaluation of depends upon an equation of a more advanced character than the ordinary linear equation under consideration. It is, however, to be noted that any particular solution, and not necessarily the general solution of the partial differential equation is sufficient to furnish an integrating factor. Moreover, in many particular cases, the partial differential equation has an obvious solution which gives the required integrating factor.

As an instance, suppose that is a function of alone, then

It is therefore necessary that the right-hand member of this equation should be independent of . When this is the case, is at once obtainable by a quadrature. Now suppose also that is unity, then must be a linear function of . The equation is therefore of the form

where and are functions of alone. The equation is therefore linear; the integrating factor, determined by the equation

is

(cf. §2.1.3).

1. The Darboux Equation

A type of equation which was investigated by Darboux is the following:²

where are polynomials in and of maximum degree .
It will be shown that when a certain number of particular solutions of the form

in which is an irreducible polynomial, are known, the equation may be integrated.

Let the general solution be

then the given equation is equivalent to

and therefore

Replace by , by , where is a third independent variable, then is a homogeneous rational function of , of degree zero, and by Euler's Theorem (§ 1.232)

Moreover satisfies the relation

in which are homogeneous polynomials in of degree .

The theory depends on the fact that if

is a solution of the given equation, is homogeneous and of degree zero, and satisfies the relation . The converse is clearly also true.

Now let

be any particular solution, where is an irreducible polynomial of degree , and let

Then, since is homogeneous and of degree ,

Also

since is a solution. This relation may be written in the form

since is a polynomial of degree and is a polynomial of degree is a polynomial of degree .

The operator has the property that if is any function of , where are themselves functions of ,

Let

be particular solutions of the given equation, where is an irreducible polynomial of degree . Let

and consider the function

where are constants to be determined. Now

where is, for every value of , a polynomial of degree . Also is a polynomial in of degree . If is to furnish the required solution when , it must be a polynomial in of degree zero, and must satisfy the relation , whence

Each polynomial contains at most terms, so that the last equation, being an identity in , is equivalent to not more than between the constants . There are, therefore, in all, at most

equations between the unknown constants . Suitable values can therefore be given to these constants if the number exceeds the number of equations, that is if

If, therefore, particular solutions are known, the general solution can be obtained without quadratures.

If and the discriminant of the equations is zero, the same result holds. Let and let the discriminant be not zero. In this case, let the constants be determined by the equations

There are now non-homogeneous equations which determine the constants . This determination of the constants gives rise to a function such that

Eliminate between these equations, then

But since is homogeneous and of degree ,

and therefore, eliminating ,

Let , then satisfies the equation

But this is precisely the condition that should be an integrating factor for the equation

If, therefore, particular solutions are known, an integrating factor can be obtained.

To return to the Jacobi equation (§ ),

In this case . The equation will have a solution of the linear form

where is a constant and . This leads to three equations between , , namely,

whence

It will be assumed that this equation has three distinct roots, , to which correspond three values of , namely, . Then

will be the general solution, when is made equal to unity, if

It is sufficient to take . The general solution is therefore

Footnotes

1. This assumption will be justified in the following chapter. ↩

2. Bull. Sc. Math. (2), 2 (1878), p. 72. ↩