# Qualitative Theory of Differential Equations

• Martin Braun
Part of the Applied Mathematical Sciences book series (AMS, volume 15)

## Abstract

In this chapter we consider the differential equation x = f(t,x)
$$x = \left( \begin{gathered} {x_1}\left( t \right) \hfill \\ \vdots \hfill \\ {x_n}\left( t \right) \hfill \\ \end{gathered} \right),an{d^{}}^{}f\left( {t,x} \right) = \left( \begin{gathered} {f_1}\left( {t,{x_1},...,{x_n}} \right) \hfill \\ \vdots \hfill \\ {f_n}\left( {t,{x_1},...,{x_n}} \right) \hfill \\ \end{gathered} \right)$$
(1)
is a nonlinear function of x1,..., xn. Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let xl(t) and x2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x1(t) and x2(t) are governed by the differential equation

## Keywords

Periodic Solution Equilibrium Point Phase Portrait Equilibrium Solution Future Time
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.