# Differential Equations and Models

## 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## Preview

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