dx

1; therefore if n+1 be odd,

the integral of the preceding equation consists of a number of terms of the form

4

n+1

4

sin cose cos 1 (30 + 2 x sin 0) fdx e

n+1

- cos X sin (≈ sin()

sin 10 ecos sin (30 + 2x sin0) fdæ e-cose X cos (v sine);

If n + 1 be even we must add the term

The factors of the operating function in this case are the same as those of the algebraical function

The quadratic factors of this expression are given by the formula

where 0

=

(1 + x)2 - 2 (1 - x) cos 20+ (1 − ≈) ̊,

(2r + 1) π

2n

From this we easily find the simple factors of the operating function to be

Therefore decomposing it into partial fractions, as in the previous Examples, we find that y consists of a number of terms of the form

π

are less than Euler, Cale. Integ. Ib.

d

dx

2

It sometimes happens that the inverse processes, such as

-1

-a X, fail, from the coefficients becoming infinite,

in the same way as the formula for integrating " fails when Thus for instance,

n =

- 1.

The method to be adopted in such cases is the same

in principle as that used for determining the value of

dx

Ꮖ

It is this: since the function becomes infinite in these cases, we so assume the arbitrary constant in the complementary function as to make the formula assume the indeterminate

form the true value of which may be easily determined

by the ordinary rules.

The assumption made with respect

to the arbitrary constant is that it shall be negative and infinite, so that the difference of the two infinite quantities may be finite.

The solution of this by the usual formula would be

To determine the real value of this, let us take the equation

dy

ay = mx,

dx

the integral of which is

Now C being an arbitrary constant, we may assume it

to be equal to

When m = a, the first term of this becomes and its

true value is easily seen, by differentiating numerator and denominator with respect to m, to be a

Therefore

when m = a.

The solution of this by the usual rule would be

This example is one of great importance, for in the application of analysis to physics, equations of this form frequently occur; and as the value of y is not simply periodic, but admits of indefinite increase, it indicates a change in the physical circumstances of the problem. Cases of this kind occur in the theory of the disturbed motions of pendulums and of the Lunar perturbations.

SECT. 2. Equations in which the coefficients are functions of the independent variable.

Equations of this class cannot be generally integrated by one method, but a considerable number may be reduced to the class discussed in the preceding section.

I. In the first place, all equations of the first order may be reduced to equations with constant coefficients by a