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But such cases are not numerous enough to warrant special notice, and their solution must be left to the ingenuity of the student. We subjoin examples requiring these and similar devices for their solution.

EXERCISES.

1. Find the difference-equations to which the following complete primitives belong.

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9.

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10.

Ux+1

sin x0 -u, sin(x+1)= cos(x-1)0-cos (3x+1)0.

Wx+1

Ux

· aux = (2x + 1) a*.

11. u2+1-2u2+1=0.

~12. (x+1)* (U2+1 − au2) = a* (x2 + 2x).

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17.

3

2

3

Ux+1 |3 — 3 a2x2 Ux+1Ux2 + 2a3x3ã‚ ̧3 = 0.

18. If P be the number of permutations of n letters taken together, repetition being allowed, but no three consecutive letters being the same, shew that

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GENERAL THEORY OF THE SOLUTIONS OF DIFFERENCE- AND
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER.

1. WE shall in this Chapter examine into the nature and relations of the various solutions of a Difference-equation of the first order, but not necessarily of the first degree, and then proceed to the solutions of the analogous Differential Equations in the hope of obtaining by this means a clearer insight into the nature and relations of the latter.

Expressing a difference-equation of the first order and nth degree in the form

-2

(Au)” +P1(Au)”1+P2(Au)”2...+P„=0.................................. (1),

1

2

PP...P being functions of the variables x and u, and then by algebraic solution reducing it to the form

(Au — p ̧) (Au — p2) ....... (▲u — p2) = 0 ....................... (2), it is evident that the complete primitive of any one of the component equations,

... (3),

▲u-p1 = 0, Au - p2 = 0... Au - Pn=0 ................

will be a complete primitive of the given equation (1) i.e. a
solution involving an arbitrary constant. And thus far there
is complete analogy with differential equations (Diff. Equa-
tions, Chap. VII. Art. 1). But here a first point of difference
arises. The complete primitives of a differential equation of
the first order, obtained by resolution of the equation with
dy
respect to
and solution of the component equations, may
without loss of generality be replaced by a single complete
primitive. (Ib. Art. 3.) Referring to the demonstration of

dx

this, the reader will see that it depends mainly upon the fact that the differential coefficient with respect to x of any function of V, V,,... V, variables supposed dependent on x, will be linear with respect to the differential coefficients of these dependent variables [Ib. (16), (17)]. But this property does not d remain if the operation A is substituted for that of and dx ;

therefore the different complete primitives of a difference(equation cannot be replaced by a single complete primitive*. On the contrary, it may be shewn that out of the complete primitives corresponding to the component equations into which the given difference-equation is supposed to be resolvable, an infinite number of other complete primitives may be evolved corresponding, not to particular component equations, but to a system of such components succeeding each other according to a determinate law of alternation as the independent variable x passes through its successive values.

Ex. Thus suppose the given equation to be

(Aux)2 — (a + x) Au2+ax = 0 .................

which is resolvable into the two equations

Aux-α=0, Aux - x=0........

........

....

· (4),

(5),

and suppose it required to obtain a complete primitive which shall satisfy the given equation (4) by satisfying the first of the component equations (5) when x is an even integer, and the second when x is an odd integer.

2

*This statement must be taken with some qualification. The reason why the primitives in question V-C1=0, V2-C=0, &c., can be replaced by the single primitive (V1- C) (V2-C)...=0 is merely that the last equation exactly expresses the facts stated by all the others (viz. that some one of the `quantities V1, V2,... is constant) and expresses no more than that. In a precisely similar way the primitives of a difference-equation of the same kind, being represented by ƒ1 (x, ux, C1)=0, ƒ1⁄2 (x, Ux, C2)=0, &c., may be equally well represented by f (x, ux, C) × ƒ2 (x, ux, C) x &c. =0. But we shall see that the latter equation must be resolved into its component equations before any conclusion is drawn as to the values of Au. It is not loss of generality that is to be feared when we combine the separate primitives into a single one, but gain. The new equation is the primitive of an equation of a far higher degree (though still of the first order), and though including the original difference-equation is by no means equivalent to it. We shall return to this point (page 184).

The condition that Au, shall be equal to a when x is even, and to x when x is odd, is satisfied if we assume

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and it will be found that this value of u, satisfies the given equation in the manner prescribed. Moreover, it is a complete primitive*.

2. It will be observed that the same values of Au, may recur in any order. Further illustration than is afforded by Ex. 1 is not needed. Indeed, what is of chief importance to be noted is not the method of solution, which might be varied, but the nature of the connexion of the derived complete primitives with the complete primitives of the component equa

* To extend this method of solution to any proposed equation and to any proposed case, it is only necessary to express Au as a linear function of the particular values which it is intended that it should receive, each such value being multiplied by a coefficient which has the property of becoming equal to unity for the values of x for which that term becomes the equivalent of Au, and to 0 for all other values. The forms of the coefficients may be determined by the following well-known proposition in the Theory of Equations.

PROP. If a, ẞ, y, ... be the several nth roots of unity, then, x being an ax +8x+zx... integer, the function is equal to unity if x be equal to n or a multiple of n, and is equal to 0 if x be not a multiple of n.

n

Hence, if it be required to form such an expression for Aux as shall assume the particular values P1, P2, ... Pn in succession for the values x=1, x=2,...x=n, and again, for the values x=n+1, x=n+2,...x=2n, and so on, ad inf., it suffices to assume

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a, B, y,...being as above the different nth roots of unity. The equation (6) must then be integrated,

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