476 16. THE CALCULUSIt appeared at a very early stage that finding an integrating factor is not in
general possible, and both Newton and Leibniz were led to the use of infinite series
with undetermined coefficients to solve such equations. Later, Maclaurin, was to
warn against too hasty recourse to infinite series, saying that certain integrals could
be better expressed geometrically as the arc lengths of various curves. But the idea
of replacing a differential equation by a system of algebraic equations was very
attractive. The earliest examples of series solutions were cited by Feigenbaum
(1994). In his Fluxions, which was written in 1671 and left unpublished during
his lifetime (see Whiteside, 1967, Vol. 3, p. 99), Newton considered the linear
differential equation that we would now write as
^ = 1-3.T + I^2 + (l + x)y.
axNewton wrote it as n/m = 1 - 3. ̧ + y + xx + xy and found that
9 I-* lj f^ ffi
» = + 3 X "6* +30^-45*Similarly, in a paper published in the Acta eruditorum in 1693 (Gerhardt, 1971,
Vol. 5, p. 287), Leibniz studied the differential equations for the logarithm and the
arcsine in order to obtain what we now call the Maclaurin series of the logarithm,
exponential, and sine functions. For example, he considered the equation a^2 dy^2 =
a^2 dx^2 + x^2 dy^2 and assumed that ÷ = by + q/^3 + ey^5 + fy^7 -\ , thereby obtaining
the scries that represents the function ÷ = asm(y/a). Neither Newton nor Leibniz
mentioned that the coefficients in these series were the derivatives of the functions
represented by the series divided by the corresponding factorials. However, that
realization came to Johann Bernoulli very soon after the publication of Leibniz'
work. In a letter to Leibniz dated September 2, 1694 (Gerhardt, 1971, Vol. 3/1,
p. 350), Bernoulli described essentially what we now call the Taylor series of a
function. In the course of this description, he gave in passing what became a
standard definition of a function, saying, "I take ç to be a quantity formed in an
arbitrary manner from variables and constants." Leibniz had used this word as
early as 1673, and in an article in the 1694 Acta eruditorum had defined a function
to be "the portion of a line cut off by lines drawn using only a fixed point and a
given point lying on a curved line." As Leibniz said, a given curve defines a number
of functions: its abscissas, its ordinates, its subtangents, and so on. The problem
that differential equations solve is to reconstruct the curve given the ratio between
two of these functions.
In classical terms, the solution of a differential equation is a function or family of
functions. Given that fact, the ways in which a function can be presented become
an important issue. With the modern definition of a function and the familiar
notation, one might easily forget that in order to apply the theory of functions it is
necessary to deal with particular functions, and these must be presented somehow.
Bernoulli's description addresses that issue, although it leaves open the question of
what methods of combining variables and constants are legal.
A digression on time. The Taylor series of a given function can be generated know-
ing the values of the function over any interval of the independent variable, no
matter how short. Thus, a quantity represented by such a series is determined for
all values of the independent variable when the values are given on any interval