ish 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.

Another simple form is

dy/dx + yP = Q,

where P, Q are functions of x only; this is known as the linear
equation, since it contains y and dy/dx only to the first degree. If

fPdx = u, we clearly have

d /dy \
--(ye^u) =e^u ( -- + Py) = e^u Q,
dx \dx /

so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
possible solution, A being an arbitrary constant. The rule for the
solution of the linear equation is thus to multiply the equation by
e^u, where u = fPdx.

A third simple and important form is that denoted by

y = px + f(p),

where p is an abbreviation for dy/dx; this is known as Clairau