1550078481-Ordinary_Differential_Equations__Roberts_

4 Ordinary Differential Equations

Sometime b etween 1673 and 1676 Leibniz invented his calculus. In 1675

he introduced the modern integral sign, an elongated letter S to denote the

first letter of the Latin word summa (sum). After some trial and error in

selecting nota tion, Leibniz settled on dx and dy for small differences in x and

y. He first used these two notations in conjunction late in 1675 when he wrote

J y dy = y^2 /2. In 1676 , Leibniz used the term "differential equation" to de-

note a relationship between two differentials dx and dy. Thus, the branch of

mathematics which deals with equations involving differentials or derivatives

was christened. To find tangents to curves Leibniz used the calculus differ-

entialis from which we derive the phrase "differential calculus," and to find

quadratures he used the calculus summatorius or the calculus integralis from

which we derive the phrase "integral calculus."

Initially, it was b eli eved that the elementa ry functions^1 would be sufficient

for representing the solutions of differential equations arising from problems in

geometry and mechanics. So early attempts at solving differential equations

were directed toward finding expli cit so lutions or reducing the solution to a

finite number of quadratures. By the end of the seventeenth century most

of the calculus which appears in current undergraduate textbooks had been

discovered along with some more advanced topics such as the calculus of

varia tion.

Until the beginning of the nineteenth century, the central theme of differ-

ential equations was to find the general so lution of a sp ecified equation or

class of equations. However , it was b ecoming increasingly clear that solving

a differential equation by quadrature was possible in only a few exceptional

cases. Thus, the emphasis shifted to obtaining approximate solutions- in

particula r , to finding series solutions. About 1820 , the French m athematician

Augustin-Louis Cauchy (1789-1857) made the solution of the init ial value

problem y' = j(x , y); y(xo) =Yo the cornerstone in his theory of differential

equations. Prior to t h e lectures developed and presented by Cauchy at the
Paris Ecole Polytechnique in the 1820 s , no adequate discussion of differen-
tial equations as a unified topic existed. Cauchy presented the first existence
and uniqueness theorems for first-order differential equations in these lectures.
Later , he extended his theory to include a system of n first-order differential
equations in n dependent variables.

There are two fundamental subdivisions in the study of differential equa-
tions: quantitative theory and qualitative theory. The obj ect of quantitative
theory is (i) to find an explicit solution of a given differential equation or
syst em of equations, (ii) to express the solution as a finite number of quadra-

(^1) Let a b e a constant and let f ( x) and g( x ) be functions. The following operations a r e called
elementary operations: f(x) ± g(x), f(x) · g(x), f(x)/g(x), (f(x))a, af(x), logaf(x), and
T(f(x)), where Tis any trigonometric or inverse trigonometric function. ElementaT'lJ func-
tions are those functions that can be generated using constants, t he independ ent variable ,
and a finite number of elementary o perations.