Introduction 5tures, or (iii) to compute a n approximate solution. Early in the development
of the subject of differential equ ations, it was thought that elementary func-
tions were sufficient for representing the solut ions of differential equations.
However, in 1725 , Daniel Bernoulli published results which showed that even
a first-order, ordinary differential equation does not necessarily have a solu-
tion which is finitely expressible in terms of elementary functions. And in the
1880 s, P icard proved that the general linear differential equation of order n is
not integrable by quadratures. In a series of papers published between 1880
and 1886, Henri Poincare (1854-1912) initiated the qualitative theory of dif-
ferential equations. The object of this theory is to obtain information about
an entire set of solut ions without actually solving the differential equation or
system of equations. For example, one tries to determine the behavior of a
solution with respect to that of one of its neighbors. That is, one wants to
know whether or not a solution v(t) which is "near" another solution w(t) at
time t = t 0 remains "near" w(t) for all t;::: t 0.
1.2 Definitions and Terminology
One purpose of this section is to discuss the meaning of the statement:"Solve the differential equation y" + y = O."
Several questions must be asked, discussed, and answered before we can fully
understand the meaning of the statement above. Some of those questions are
"What is a differential equation?""What is a solution of a differential equation?""Given a particular differential equation and some appropriate constraints,
how do we know if there is a solution?"
"Given a particular differential equation and some appropriate constraints,
how do we know how many solutions there are?"
"How do we find a so lution to a differential equation?"We will answer the first two questions in this section and devote much of the
remainder of the text to answering the last three questions.
At this point in your study of mathematics you probably completely un-
derstand the meaning of the statement:
"Solve the equation 2x^4 - 3x^3 - 13x^2 + 37x - 15 = O."
You recognize this is a n algebraic equ ation. More specifically you recognize
this is a polynomial of degree four. Furthermore, you know there are exactly