Let k be a positive constant. Which of the following is a logistic differential equation?

Let k be a positive constant. Which of the following is a logistic differential equation?
(A) dy/dt=kt
(B) dy/dt=ky
(C) dy/dt=kt(1 -t)
(D) dy/dt=ky(1-t)

(E) dy/dt=ky(1-y)

Please show details.

Suppose that y(t) describes the quantity of a population at time t. For example, y(t) could be the number of milligrams of bacteria in a particular beaker for a biology experiment, or y(t) could be the number of people in a particular country at a time t. A model of population growth tells rules for how such a population changes over time. The simplest model of population growth is the exponential model, which assumes that there is a constant parameter k, called the growth parameter, such that y'(t) = k y(t) holds for all time t. This differential equation itself might be called the exponential differential equation .

An alternative model to allow for the fact that there are limits to growth in all known biological systems. It is now called the logistic differential equation.

The equation involves two positive parameters. The first parameter k is again called the growth parameter and plays a role similar to that of in the exponential differential equation. The second parameter K is called the carrying capacity. The logistic differential equation is written

dy/dt = ky [1 - y / K]

or dy/dt = ky [1 - y], when the carrying capacity K is taken 1.

Now when y(t) is very small, then y(t)/K is close to 0, so the entire factor [1-y(t)/K] is close to 1 and the equation itself is then close to y'(t) = ky(t); we then expect that the population grows approximately at an exponential rate when the population is small. On the other hand, if y(t) gets to be near K, then y(t)/K will be approximately 1, so [1-y(t)/K] will be approximately 0, and the logistic differential equation will then say approximately y'(t) = ky(t) 0 = 0. The growth rate will be essentially 0, so the population will not grow significantly more.

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来源： https://www.cnblogs.com/coursehero/p/16150396.html