Differential Equations and Their Applications pp 493-637 | Cite as

# Qualitative Theory of Differential Equations

Chapter

## Abstract

In this chapter we consider the differential equation x = f(t,x)
is a nonlinear function of 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)

_{1},..., x_{n}. 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 x_{l}(t) and x_{2}(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 x_{1}(t) and x_{2}(t) are governed by the differential equation## Keywords

Periodic Solution Equilibrium Point Phase Portrait Equilibrium Solution Future Time## Preview

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## Copyright information

© Springer Science+Business Media New York 1975