Differential Equations and Their Applications pp 280-299 | Cite as

# Qualitative theory of differential equations

Chapter

- 597 Downloads

## Abstract

In this chapter we consider the differential equation where and is a nonlinear function of

$${\rm{\dot x}}\, = \,{\rm{f}}\,\left( {t,{\rm{x}}} \right)$$

(1)

$${\rm{x}}\,{\rm{ = }}\,\left( {\begin{array}{*{20}{c}}
{{x_1}\left( t \right)}\\
\vdots \\
{{x_n}\left( t \right)}
\end{array}} \right),$$

$${\rm{f}}\,\left( {t,\,{\rm{x}}} \right)\, = \,\left( {\begin{array}{*{20}{c}}
{{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)}\\
\vdots \\
{{f_n}\,\left( {t,{x_1}, \ldots ,{x_n}} \right)}
\end{array}} \right)$$

*x*_{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*_{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 (1). 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.## Keywords

Loss Rate Equilibrium Solution Qualitative Theory Reinforcement Rate Solution Curve
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Coleman, C. S., Combat Models, MAA Workshop on Modules in Applied Math, Cornell University, Aug. 1976.Google Scholar
- 2.Engel, J. H., A verification of Lanchester’s law,
*Operations Research*,**2**, (1954), 163–171.CrossRefGoogle Scholar - 3.Howes, D. R., and Thrall, R. M., A theory of ideal linear weights for heterogeneous combat forces,
*Naval Research Logistics Quarterly*, vol. 20, 1973, pp. 645–659.CrossRefGoogle Scholar - 4.Lanchester, F. W.,
*Aircraft in Warfare, the Dawn of the Fourth Arm*. Tiptree, Constable and Co., Ltd., 1916.Google Scholar - 5.Morehouse, C. P.,
*The Iwo Jima Operation*, USMCR, Historical Division, Hdqr. USMC, 1946.Google Scholar - 6.Newcomb, R. F.,
*Iwo Jima*. New York: Holt, Rinehart, and Winston, 1965.Google Scholar

## Copyright information

© Springer-Verlag, New York Inc. 1978