416 Ordinary Differential Equationsr s E=CD-AB Possible Arms Race(s)
(^0 0) +
0 0
6. What kinds of arms races can develop if r < 0, s < 0 and lines L1 and
L 2 are parallel (C/A = B/D; E = O)? (HINT: Consider two separate
cases, where L 1 is above L 2 and vice versa.)10.2 Phase-Plane Portraits
There are two fundamental subdivisions in the study of differential equa-
tions: (1) quantitative theory and (2) qualitative theory. The object of quanti-
tative theory is (i) to find an explicit solution of a given differential equation or
system of equations, (ii) to express the solution as a finite number of quadra-
tures, or (iii) to compute an approximate solution. At an early stage in the
development of the subject of differential equations, it appears to have been
believed that elementary functions were sufficient for representing the solu-
tions of differential equations which evolved from problems in geometry and
mechanics. However, in 1725 , Daniel Bernoulli published results concerning
Riccati's equation which showed that even a first-order ordinary differential
equation does not necessarily have a solution which is finitely expressible in
terms of elementary functions. In the 1880s, Picard proved that the gen-
eral linear differential equation of order n is not integrable by quadratures.
At about the same time in a series of papers published between 1880 and
1886, Henri Poincare (1854-1912) initiated the qualitative theory of differ-
ential equations. The object of this theory is to obtain information about
an entire set of solutions without actually so lving 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 = to remains "near" w(t) for all t ;::: t 0 for which both v(t) and w(t)
are defined.