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),

y = f (a)

1

...+fe-xxdx; (13)

an expression which involves n signs of integration, and consequently n arbitrary constants; and, if these are introduced, the

result becomes

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,

where k and y are two new undetermined constants; and if

(15)

24 Je-" cos br x dr

ax sin bxx dx;

eax
2 ea2 (L. sin bx - cosba) fe-ar

L2+ M2

si

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,

f (*)a
dx

if a1 = a2 = ... = a; then, according to Art. 21, if y(x) is equal to the reciprocal of p(x), where (r) is the product of all the factors of ƒ (short of the equal factors,

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
dx3

d

dx

dx

..

and a constant must be added at each integration.

Let x=0; y = c12 ea √ = 1 x + C2e-a√=1x

473.] The preceding process, it will be observed, involves operations represented by symbols of the general forms

where r 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

sive terms of the expansion: but (

in either of the following forms,

+ a) may be expressed