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Merca) dre:*** Y cos (.r sine);

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305 and the factors of cos () are therefore (-4) (1 + i)(-10) (1 + %), &c.

O

.

ar

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the The factores precio y la mention on this case are

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

* {(1 + x)" + (1 – X)"}.

The quadratic factors of this expression are given by the formula

(1 + x)” – 2 (1 – x) cos 20 + (1 - %)',

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

+(-)} tan 0.

Therefore decomposing it into partial fractions, as in the previous Examples, we find that y consists of a number of terms of the form .2 (cos 6)n-1, sin (x tan 6) sdx X cos (x tan 6)

n - cos (x tan 8) Sdx X sin (x tan e)s

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O receiving the values , ; , &c., so long as they are less than Euler, Calc. Integ. Ib.

It sometimes happens that the inverse processes, such as

-a) X, fail, from the coefficients becoming infinite, in the same way as the formula for integrating r" fails when n = -1. Thus for instance,

( -a) cm = 40 when m = a.

ma

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- a

The method to be adopted in such cases is the same in principle as that used for determining the value of *. 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

fono

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.

(21) Let the equation be

and - ay = fa? The solution of this by the usual formula would be

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

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Now C being an arbitrary constant, we may assume it to be equal to

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so that

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- a

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 when m = a. Therefore

y = x 6"* + Ceas is the solution of the equation

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The solution of this by the usual rule would be

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This is easily seen to be

æ sin nr

2n so that the solution of the given equation is

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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.

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

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