国际数学建模竞赛的选拔题！来试试看啊？

Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, a technological innovation, or a cultural fad amongst a population. The members of the population can be divided into two classes: those who have the information and those who do not. In a fixed population whose size is known, it is reasonable to assume that the rate of diffusion of population is proportional to the number who has the information times the number yet to receive it. If X denotes the number of individuals who have the information in a population of N people, then a mathematical model for social diffusion is given by dX/dt=kX(N-X), where t represents time and k is a positive constant.

1.Solve the differential equation above and try to graph the function you solved.
2.At what time is the information spreading fastest?
3.How many people will eventually receive the information?

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