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Math Help - The Logistic Equation

  1. #1
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    The Logistic Equation

    Hello,
    I was wondering if anyone could help me with my logistic equation problem; the question asks me what logistic equation is solved by y=2/(1+e^-t).
    If I take the derivative from above I get 2et/(e^t+1)^2, however I have no idea how to get it to the form y=cy-by^2. I know it has something to do with finding the residue and splitting the variables, but I am stuck.
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  2. #2
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    You can solve the differential equation y' = cy - by^2, and write the solution in terms of generic c and b.
    Then see how you should choose c and b to get y=2/(1+e^-t).
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  3. #3
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    I solved y'=cy-by^2 to get y=c/(b+(1/C)e^(-ct)) but I'm not sure how I'm supposed to solve this for the correct logistic equation.
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  4. #4
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    What values should you pick for c and b to get the solution into the form you are looking for 2/(1+e^-t)?
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  5. #5
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    Okay, I've rearanged it so that Ce^ct=2/(c(1+e^-t)-2b)) but I don't know where to go from there. I think I have to take the ln() of the left side and multiply it by 1/c, but I'm not sure where to go from there. Sorry I'm so slow, but this is strange as this question is done backwards from what we were taught.
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  6. #6
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    The key thing is to notice for
    c/(b+(1/C)e^(-ct)) = 2/(1+e^-t)

    We must have e^(-ct) = e^-t. What is c?
    Now can you solve for b?
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