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Math Help - Differential Equations

  1. #1
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    Differential Equations

    Suppose that the population P(t) of a country satisfies the differential equation

    dP/dt=kP(200-P)

    with k constant. Its population in 1940 was 100 million and and was then growing at the rate of 1 million per year. Predict this country's population for the year 2009.
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  2. #2
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    Quote Originally Posted by cowracer3 View Post
    Suppose that the population P(t) of a country satisfies the differential equation

    dP/dt=kP(200-P)

    with k constant. Its population in 1940 was 100 million and and was then growing at the rate of 1 million per year. Predict this country's population for the year 2009.

    After separating variables,and taking into account the facts given by the problem ,we integrate from 100 to p the fraction :


    ................... \frac{dt}{p(200-p)}....................

    to get : \frac{ 1}{200}\ln\frac{ p}{200-p}.................................................. .............................1

    Then we integrate ,kdt, from 1940 to 2009 and put k =1 and we get:

    .............................69................... .............................................2


    Now we equate (1) to (2) (1)=(2) we get:


    ................... \frac{ p}{200-p}=e^13800=c.....................3


    After doing calculations we have:



    ...................  p=\frac{200c}{1+c}.................................................. ..........................4

    And since c is very large we can substitute 1+c by c in 4 and thus finally have:

    .....................................p=200 million........................................... .................................
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