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6(a* +1 – 1)

a -1·,·u,+1...+η -1+ const. ; α*

1

1 =const.

(α-1-1) μ.μ.-1...+1-2 (Τr. L. App. 373-5; L. C. D. 955-60.)

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n

+(-1)" {E}(4,"Uz. S. Vr+n) + .87-(4,"u..E+10+o)

1

a

n(n + 1)... (n + p - 2) + &c. +

,(Aru,. En+-10:+r)}. 1

(r-1) (Tr. L. App. 876, 7; L. C. D. 959–62.)

2

.

4

1.2.3

(24.) Eu: =S; u.- fu,+ d-U, +6 dzu,

+ &c.
=(ex - 1)-u,. B
"u,=(e :- 1) -"u,,'y
,(b.u.)=b*(5.6"1)-u,. 6
(6*.0,)=b*(b.edi - 1) -n.ur.

(TP. L. App. 378; L. C. D. 963–80.)

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EQUATIONS OF DIFFERENCES. (25.) The complete integral of an equation of differences of the nth order

0=(@, U7, U, +1, &c. Uz+n) must contain n independent arbitrary constants. (26.) The general equation of the first order and degree is

U:+1=2,3, + bx. Let u.=v..07.0...-=v. Pow), then (Os+.-P..)by, or 4,6..P..),

b .. u=Pm(am). E,

}

+ const.

Pmlam)

(TT. L. App. 879_81; L. C. D. 1038.) (27.) The general equation of the nth order and first degree is

Urte + 'aUz+r--+ *ax-Uz tm-2 + &c. + "az.uz=b; (1) the integration of which is reducible to that of the equation

Uz+n + 'ax.Uz+n-1 + ac.Uz+n-2+ &c. + "az.u,=0: (2) which latter equation is always integrable if n-1 particular integrals can be obtained; and if m particular integrals can be obtained, it may be reduced to an equation of the (n-m) order. The general method may, to avoid complexity, be illustrated by its application to the equation

Ug+3 + '&c.Uz +2 + $ac. Uz +1 + 3a,U;=bz. Let this equation be obtained by elimination from the equations

Uz+1+ loc. Uz = 'ur,
"Uz +1 + Pvcu,= 'uz) (3)

+1 + $07.*uz = bx

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* See Appendix.

30.

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30x

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u=P(~ Pom){*C+2,
P

+C

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Ex I

ĮP, Om. P-vm) P(-30,

then laz='0x+&+ +1 + 3v.rs

*a;="vx+1.%0x+1+ to 30. + logo
3ax='oxvx.Sv

C+ Σ
+ Es
Pm(-'um)

(5)
P-(-,
-30

m) in which 'C, °C, and °C are arbitrary constants. From these equations we may obtain by elimination

Un='C. U. +C.U, +C.SU, + V,, in which 'U.=P~(~ 'vm),

.
.-,

(6)
P.(-.)
P

-30
%

~~ Pry P

2P

;

4-1

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m(

2 1

If br=0, then V2=0, and the equation

U,='C.’U. +°C.'U, + 3C. U. is the complete integral of

U2+3+ 'a .Ux+2+ a,.Ur+1+302.0,=0.

If this equation is known, then 10,3 %v,, &c. may be determined, for from (6)

U. 10,=

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x + 1

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&c. = &c. these quantities being substituted in V, and the result added to the complete integral of (2), the complete integral of (1) will be obtained, If n-1 of the quantities U, are known, the more may

be obtained from the equations (3). (28.) If the coefficients of (2) are constant, that equation will be satisfied by assuming u,=e"; then, dividing by e', we have

e" + 1a.e"-1 + 'a.en-2 + &c. + "-la.e+ "a=0. Let e,, e,, &c. en, be the roots of this equation, then

Ur='C.e* + PC.cm+ &c. + "C.em is the complete integral required.

If

any of the quantities ez, e,, &c. are equal, or impossible, the changes which take place in the value of Uz are analogous to those in (Int. Calc. 43.) (29.) If we wish to extend the integral of (2) to (1), then

6 Uz='C.e* + &c. + "C.en +

1+ 'a + a + &c. + "a

(Tr. L. App. 382–5; L. C. D. 103652.) (30.) An equation of the second order and first degree

Uz++'ag.uzti+Paz.u = , (1) may be thus solved : assume

1-2

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Un=v7.P(-'an),

and Cr+=

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