Abstract
In this chapter we consider the differential equation
where
and
is a nonlinear function of x 1, ..., x n. 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 x 1(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. In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time t. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.
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Braun, M. (1993). Qualitative theory of differential equations. In: Differential Equations and Their Applications. Texts in Applied Mathematics, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4360-1_4
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DOI: https://doi.org/10.1007/978-1-4612-4360-1_4
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