This will be done in a precisely similar way:

17. In a short note in Tortolini's Annali (Series 1. vol. v.) Maonardi gives a solution of the linear difference-equation with constant coefficients that does not require the preliminary solution of the algebraical equation for E, but the results do not seem of much value.

3. Ux+2+2x+1 + u2 = x (x − 1) (x − 2) + x (− 1)*.

4. u2+2-2mu 2+1 + (m2 + n3) u ̧=m*. Ихад 2mu2+1

5. Aux+A3ux=x+ sin x.

11. A person finds his professional income, which for the the first year was £a, increase in A. P., the common difference

1

being £b. He saves every year of his income from all

m.

sources, laying it out at the end of each year at r per cent. per annum. What will be his income when he has been x years in practice?

12. A seed is planted-when one year old it produces ten-fold, and when two years old and upwards eighteen-fold. Every seed is planted as soon as produced. Find the number of grains at the end of the ath year.

1. SINCE no class of equations of an order higher than the first have been solved with the completeness which marks the solution of linear difference-equations with constant coefficients, it becomes very important to find what forms of equations can be reduced to this class. The most general case of this reduction is with regard to equations of the form

Ux+n

+ A1$ (x) Ux+n−1 + a ̧‡ (x) † (x − 1) Ux+n-2

=

+ A ̧‡ (x) † (x − 1) † (x − 2) Ux+n-3 + &c. X......(1), where A, A,... A, are constant, and (x) a known function. These may be reduced to equations with constant coefficients by assuming

For this substitution gives

Ux+n = $ (x) & (x − 1) † (x − 2) ... P (1) vx.

Ux+n_1 = (x − 1) p (x − 2) ... ø (1) Vx+n-17

(2).

and so on; whence substituting and dividing by the common factor (x) (x − 1) ... 6 (1), we get,

In effecting the above transformation we have supposed to admit of a system of positive integral values. The general transformation would obviously be

u2=$(x−n) $ (x − n − 1) ... $ (r),

r being any particular value of x assumed as initial. Equations of the form

2x

Ux+n+ A1α*Ux+n−1 + Â ̧àa¤μx+n_2 + &c. = X,

are virtually included in the above class. For, assuming (x) = a*, they may be presented in the form

Ux+n

+4 ̧$ (x) Ux+n-1 +a‚α ☀ (x) † (x − 1) Ux+n-2+ &c. = X.

1

Hence, to integrate them it is only necessary to assume

2. By means of the proposition in the last article we can solve all linear binomial equations. Let the equation be

Take logarithms of both sides and let log vxn+1=Wx, then we have

Wx+n−1 +Wx+n_2 + &c. + w2 = log A, .............................. (8),

a linear difference-equation with constant coefficients. Solving this we obtain w, and thence v., which enables us to put (6) into the form

by Art. 1, and thus the equation is solved.

(9)

Such equations are however substantially equations of the first degree, and should be treated as such. They state a

connection between consecutive members of the series urg urin, Urin &c., and leave these last wholly unconnected with intermediate values of u. We should therefore assume x = ny and the equation would become a linear difference-equation of the first order, the independent variable now proceeding by unit increments.

3. Equations of the form

Ux+1 Ux + ɑx2x+1 + bxUx=Cx

..(10)

can be reduced to linear equations of the second order, and, under certain conditions, to linear equations with constant coefficients*. Assume

Then for the first two terms of the proposed equation, we have

a linear equation whose coefficients will be constant if the functions ba+1 and abx + c2 are constant, and which again by the previous section may be reduced to an equation with constant coefficients if those functions are of the respective forms

Ap (x), Bp (x) ¤ (x − 1).

4. Although linear difference-equations with variable coefficients cannot generally be solved, yet, in virtue of their

Should c, be zero the equation is at once reduced to a linear equation of the first order by dividing by uxux+1, and taking as our new dependent: variable.

Их