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Differential Equations and Models

  • J. David LoganEmail author
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

In science, engineering, economics, and in most areas having a quantitative component, we are interested in describing how systems evolve in time, that is, in describing a system’s dynamics. In the simplest one-dimensional case the state of a system at any time t is denoted by a function, which we generically write as u = u(t). We think of the dependent variable u as the state variable of a system that is varying with time t, which is the independent variable. Thus, knowing u = u(t) is tantamount to knowing what state the system is in at time t. For example, u(t) could be the population of an animal species in an ecosystem, the concentration of a chemical substance in the blood, the number of infected individuals in a flu epidemic, the current in an electrical circuit, the speed of a spacecraft, the mass of a decaying isotope, or the monthly sales of an advertised item. Knowledge of u(t) for a given system tells us exactly how the state of the system is changing in time. Figure 1.1 shows a time series plot of a generic state function. We use the variable u for a generic state; but if the state is “population”, then we may use p or N; if the state is voltage, we may use V . For mechanical systems we often use x = x(t) for the position.

Keywords

Bifurcation Diagram Arbitrary Constant Equilibrium Solution Erential Equation Computer Algebra System 
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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Copyright information

© Springer Science+Business Media,LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska—LincolnLincolnUSA

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