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.
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© 2010 Springer Science+Business Media,LLC
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Logan, J.D. (2010). Differential Equations and Models. In: A First Course in Differential Equations. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7592-8_1
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DOI: https://doi.org/10.1007/978-1-4419-7592-8_1
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