# Qualitative theory of differential equations

• Martin Braun
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 11)

## Abstract

In this chapter we consider the differential equation
$$\dot x = f\left( {t,x} \right)$$
(1)
where
$$x = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)} \\ \vdots\\ {{x_n}\left( t \right)} \end{array}} \right],$$
and
$$f\left( {t,x} \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \\ \vdots\\ {{f_n}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \end{array}} \right]$$
is a nonlinear function of x1, ..., xn. Unfortunately, there are no known methods of solving Equation. This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of explicitly. For example, let x1(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. In this case, we are not really interested in the values of x1(t) and x2(t) at every time t. Rather, we are interested in the qualitative properties of x1(t) and x2(t). Specically, we wish to answer the following questions.

## Keywords

Equilibrium Point Phase Portrait Bifurcation Point 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.

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