Exponential GrowthIn this section we discuss the differential equationdydt k y;where k is a positive constant. In words, this equation says that the rate at which thevariable y changes is proportional to the value of y; thus the larger y becomes, themore quickly it increases. The result is that y grows, slowly at first and then veryquickly, a phenomenon known as exponential growth.The Exponentia
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Exponential Growth
In this section we discuss the differential equation
dy
dt k y;
where k is a positive constant. In words, this equation says that the rate at which the
variable y changes is proportional to the value of y; thus the larger y becomes, the
more quickly it increases. The result is that y grows, slowly at first and then very
quickly, a phenomenon known as exponential growth.
The Exponential Growth Equation
The exponential growth equation is the differential equation
dy
dt k y (k > 0):
Its solutions are exponential functions of the form
y y0 ekt
where y0 y(0) is the initial value of y.
Figure 1 shows the graph of a typical exponential function, assuming y0 > 0
and k > 0. Because of the factor of et, an exponential function increases quite quickly
a Figure 1: Exponential growth.
as t increases, as illustrated in Figure 2
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