A (x+c) e2ax √=1 2 +▲ cos (ax − a); where a and a are arbitrary constants. 472.] The integral value of the right-hand member of (4) may also be determined in the following manner. The function in terms of the operating symbols is evidently a rational fraction in powers of d ; it may consequently be decomposed into a series dri dx of simple fractions by the process explained in Chap. II, Section 2, of the present volume. Firstly, let us suppose all the roots to be unequal; then by Therefore, introducing the subject of the operating symbols, And consequently by reason of the theorem cited in (5), an expression which involves n signs of integration, and consequently n arbitrary constants; and, if these are introduced, the which result is identical with (106), Art. 451. If there is one pair, or are many pairs, of imaginary roots, we may transform the expression by the process of Article 452. Thus if a, and a, are a pair of conjugate imaginary roots, then ai = a+b√−1, a = a-b√ -1; C¡ea¡¤+C;ea;* = eax {C¡eb√=1x + C;e−b√=1x} =eax k cos (bx+y) ; where k and y are two new undetermined constants; and if (15) Je-az cos bxx dx 2 ea2 (L sin ba — M cosba) fe-az sin bæ x dr; L2 + M2 and this again may be further simplified if L and м are replaced by r cose and r sin respectively. d If however m roots of ƒ (1) are equal to each other; that is, dx if a1 = α2 = ... = a; then, according to Art. 21, if y(x) is equal to the reciprocal of p(x), where p(x) is the product of all the factors of ƒ (d) short of the equal factors, dx to all these terms let the subject x be affixed; then since and as the constants introduced by integration are arbitrary, in the first m terms, m and only m constants will be brought in, and the remaining n-m constants will arise in the other integrations. If the roots corresponding to the sets of equal factors are imaginary, the process of integration is the same; the result however is so complicated that it is not worth while to express it at length. Ex. 1. -6a+11ady-6a'y = x2. d3y d dx dx .. = e2ax {C′x+c′′} + (4 a2 — n2) sin nx + 4 an cos nx (n2 + 4a2)2 2a dx2, = 2a√-1 and a constant must be added at each integration. 473.] The preceding process, it will be observed, involves operations represented by symbols of the general forms where is unity or some other positive and integral number; and as the operation which such a symbol represents is subject to the laws of distribution and of repetition, we may expand the operative symbol, and operate on x with the several and succes d sive terms of the expansion: but (+a) may be expressed in either of the following forms, r(r+1) a2 (d) |