_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - 1/2[Delta]^2u0 + 19/60 [Delta]^3u0 - ...)
h _/x0
+ 1/12 ([Delta]'u_n - 1/2[Delta]'^2u_n + 19/60 [Delta]'^3u_n - ...),
where accented differences denote that the values of u are read
backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in S
10) u_n - u_n-1.
(iii.) Expressed in terms of central differences this becomes
_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]^3u_n - ...
h _/x0
+ 1/12 [mu][delta]u0 - 11/720 [mu][delta]^3u0 + ...
/ 1 11 191 2497 \ / \
= [mu]([sigma] - -- [delta] + --- [delta]^3 - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
\ 12 720 60480 3628800 / \ /
(iv.) There are variants of these formulae, due to taking hum+1/2 as the
first approximation to the area of the curve between um and um+1; the
formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 := [sigma](u_n -
u0) (see MENSURATION).
20. The formulae in the last section can be obtained by symbolical
methods from the relation
_
1 / 1 1
- | udx = - D^1 u = --.u.
h _/ h hD
Thus for central differences, if we write [theta] := 1/2hD, we have [mu]
= cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
the result in (iii.) corresponds to the formula
/ / 1 2 2.4 \
sinh [theta] = [theta] cosh [theta]/ (1 + - sinh^2[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
/ \ 3 3.5 3.5.7 /
REFERENCES.--There is no recent English work on the theory of finite
differences as a whole. G. Boole's _Finite Differences_ (1st ed.,
1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
treatise, in which symbolical methods are employed very early. A. A.
Markoff's _Differenzenrechnung_ (German trans., 1896) contains general
formulae. (Both these works ignore central differences.) _Encycl. der
math. Wiss._ vol. i. pt. 2, pp. 919-935, may also be consulted. An
elementary treatment of the subject will be found in many text-books,
e.g. G. Chrystal's _Algebra_ (pt. 2, ch. xxxi.). A. W. Sunderland,
_Notes on Finite Differences_ (1885), is intended for actuarial
students. Various central-difference formulae with references are
given in _Proc. Lond. Math. Soc._ xxxi. pp. 449-488. For other
references see INTERPOLATION. (W. F. SH.)
DIFFERENTIAL EQUATION, in mathematics, a relation between one or more
functions and their differential coefficients. The subject is treated
here in two parts: (1) an elementary introduction dealing with the more
commonly recognized types of differential equations which can be solved
by rule; and (2) the general theory.
_Part I.--Elementary Introduction._
Of equations involving only one independent variable, x (known as
_ordinary_ differential equations), and one dependent variable, y, and
containing only the first differential coefficient dy/dx (and
therefore said to be of the first _order_), the simplest form is that
reducible to the type
dy/dx = f(x)/F(y),
leading to the result fF(y)dy - ff(x)dx = A, where A is an arbitrary
constant; this result is said to solve the differential equation, the
problem of evaluating the integrals belonging to the integral
calculus.