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and as these conditions must consist with equation (79), we have after substitution

(97)

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but each term of the series comprehended within the symbol of aggregation vanishes, because a, a,... a, are the n roots of the characteristic equation; and therefore we have

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ƒ′(a) = (a—a2) (a—a ̧)....... (a—aŋ)+(a—a1) (a−α ̧).....(a—a„)

+ ... + (a−a1) (a—a2).......(a—a‚—1); (101)

.. ƒ(a1) = (α1—α) (α-α)... (α-a,);

f(a2) = (α2-a1) (a2 —αz)..... (α2—an);

-

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(102)

Of these equations let us take the first to be the type: it is plainly of n-1 dimensions in a,, so that

n-1

2-3

ƒ(a1) = a12"-1+c112"−2+ C2α12"-3 + ... + Cm-291+Cn-1, (103) where C1, C2,... C-1 are functions of a, a,,...a; and let us multiply equations (100) severally by Cn-1, Cn-2,... C1, 1 and add them then the coefficient of A, is f(a), and the coefficients of Ag, Ag,... A vanish, because (103) vanishes by virtue of the first λη, of (102) when a1 is replaced by a, or a... or a,; and therefore ultimately we have λ(a) = 1; so that

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and including the constant factor in the arbitrary constant c we have

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452.] This is the general integral of the differential equation (79), when all the roots of the characteristic are unequal. And if x = 0, that is, if the right-hand member of (79) = 0, then

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(107) an expression which is easily verified by means of substitution in (79), and each of the terms of which is a particular integral; and as all are different, n different arbitrary constants are contained in it, and the integral is therefore general; and the form of (106) indicates that the general integral is the sum of n particular integrals, each of which involves or may involve a different arbitrary constant.

If there are pairs of impossible roots in the characteristic of (79) they enter as conjugates: suppose a pair to be a¡, aj: so that

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if c+c, k cos y, (c;-c;) √-1=-ksiny; and where of course k and y are possible quantities. In the case therefore of a pair of imaginary roots, two terms of (107) will in combination produce a trigonometrical function of the form (108), and instead of the arbitrary constants c; and c, we have the new constants, equally arbitrary, k and y. And a similar process of combination is also applicable to the latter unintegrated terms of the general expression (106).

453.] The following are examples of the preceding process of integration;

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- (6a+0')=0,

11a2―0′′= 00',

—6a3 = 00′′ ;

whence we have

dy

dy
-6a+11a2-6ay = eTM,

y

+0′′y } − (6a+ 0) —2 + (11 a2 — 0′′) dy — 6 a3y = eTM2;

−(6a+0)

dx2

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dx

dx+on = emx;

03+6a02+11a20+6a3 = 0;

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and therefore in accordance with equation (86),

f(a) = (a-a) (a-2a) (a-3a),

f'(a) = (a−2a) (a−3 a)+(a−3 a) (a−a)+(a− a) (a−2a),
ƒ′(a)=2a2, ƒ′(2a) =—a2, ƒ′(3a) = 2a2;

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454.] In the preceding investigations we have, at least tacitly, supposed all the roots of the characteristic to be unequal: for if two or more of them are equal, the value of y, as expressed in (89) and found by elimination from the group of equations (88), becomes indeterminate, and the subsequent processes of Art. 451 fail. Now, to take a particular case, let us suppose two roots to be equal, say a2 = a1; then the terms corresponding to these two roots become

C1 ea1 +C2 ea12 = (C1+C2) ea* = c'ex;

and thus the two particular integrals will introduce only one arbitrary constant, and the general integral will contain only n-1 different constants: and consequently its generality is lost. Let us return then, and suppose m roots, a1, a,...am, of the characteristic to be equal, that is,

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and, for the sake of simplicity, consider a differential equation which has no second member, observing that the generality of the process is not lost by the restriction.

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so that the roots are thus made to be unequal; but they will

become equal if i

if

...

2

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= e1* {c′+c′′x+c"" x2 + ... + C(m) xm + c(m+1) xm+1+...},

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Of these equations let us take the first m to determine the new constants c', c', ... c(); and then let us suppose i = 0, so that all the subsequent terms vanish, and the m roots of the characteristic become equal; and thus ultimately for the general integral we have

y = {c′ + c′′x + ... + c(m) x1 } ea2 + Cm + 10m+1* + ... + Crea12. c′+ ea1*+Cm+10am+1* +.

Thus if two roots of the characteristic are equal

y = {c′+c′′x} ea12 + Czeaz* + ... + C„ea2.

Also we may consider the case of equal roots in the following manner and this is perhaps more direct.

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where a is a constant and u is a function of x; then substituting

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Now this equation is satisfied if u = a constant, and ƒ (a) = 0, that is, if for a we substitute one of the roots of the characteristic: let then a be substituted for a, and c, for u, so that (110) becomes

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which is a particular integral; and in the same way may the other particular integrals, and consequently the general integral, be found. If however two roots of the characteristic are equal, saya, a1, then ƒ(a1) = 0, ƒ'(a1) = 0; and (111) is satisfied when

=

d2 u
= 0;
dx2

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Similarly if m roots of the characteristic are equal, it is necessary that

dm u
= 0;
dxm

u = c + c′′x +...+c(m) xm;

and thus we have the form of the general integral when m roots of the characteristic are equal.

455.] SECOND METHOD.-We may also apply to the solution of linear equations with constant coefficients the process of successive reduction which has been investigated in the preceding Section. Taking (80) to be the type equation, let

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where a is an undetermined constant, and u is a function of x:

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