On some models based on first order differential equations

Abstract

This paper presents some models based on first order differential equations. Such models include population growth, drug distribution in the body and dating archaeological samples models. The paper also discusses the formulations, solutions and applications of such models. Keyword: Models, Growth, Decay, Drug Distribution, and First order INTRODUCTION Over the last few decades, mathematics has broken out into a whole new range of applications in the social sciences, biology, medicine, management, e.t.c. and it seems, almost every field of human endeavor, providing qualitative, if not quantitative models where none had existed or even been contemplated before. Mathematical techniques now play an important role in planning, managerial decision-making, and economics, which has probably been longest quantified of the social sciences (c.f. Burghes etal, 1980). The underlying theme in all applications of mathematics to real situations is the process of mathematical modeling. By this we mean the method of translating a real problem from its initial context into a mathematical description, that is, the mathematical models. This mathematical problem is then solved, and the resulting mathematical solutions must be translated back into the original context. The theory of ordinary differential equations. Equations: On a more practical level, it could be claimed that the spread of modern industrial civilization, for better or for worse, is partly a result of man’s ability to solve the differential equations which govern so many of our industrial processes, be they chemical or engineering (c. f. Burghes et al, 1980) A first order ordinary differential equation is a relation between the derivative of an unknown function x (t), where t is a real variable, the function x itself, the independent variable t, and given function of t. Denoting dt dx by  . x (by convention differentiation with respect to t is denoted by a dot, and not by a prime), we assume that in some domain D we can express  x as a function of t and x , namely   x t f x , .   (1) where f is a given function on the subject D (assumed open and connected) of 2 R taking values in R . A function   t x   which when substituted in (1) reduces it to an identity for each t in some interval (a,b) is called a solution of (1) over the interval (a,b). If f is continuous on D, each solution   t x   will define a smooth curve in D called an integral curve of the equation. Through each point   x t, of D there will pass an integral curve whose gradient at that point is given by   x t f , . Which curve we choose as our solution will depend on the initial data given. This is illustrated in Fig. 1 below.