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Homework Help: Show directly that P is a solution to the differential equation

  1. Jun 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Show directly that
    [itex]P=\frac{L}{1+Ae^{-kt}}[/itex]

    is a solution to the differential equation

    dP/dt=kP(1-P/L)

    2. Relevant equations

    -

    3. The attempt at a solution

    I assume that all I need to do is differentiate P with respect to t. However, as you can see below, either I'm doing it wrong or I don't know what I need to do next.

    [itex]dP/dt=(d/dt)\frac{L}{1+Ae^{-kt}}[/itex]

    First I take 1+Ae^-kt to be 1/x, and differentiate it to -x^-2. Then I multiply it by the derivative of x, which is -Ake^-kt. So I finish up with [itex]\frac{LAe^{-kt}}{(1+Ae^{-kt})^{2}}[/itex]

    Im stuck here. I have no idea how to get the correct answer :S
     
  2. jcsd
  3. Jun 3, 2012 #2

    SammyS

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    You're missing k in that last expression:
    [itex]\displaystyle \frac{kLAe^{-kt}}{(1+Ae^{-kt})^{2}}[/itex]​

    Multiply the numerator & denominator by L .

    Change the [itex]Ae^{-kt}[/itex] in the numerator to [itex]\left(1+Ae^{-kt}\right)-1[/itex]
     
  4. Jun 3, 2012 #3
    Thanks! I'll try it out and see if I can get it.
     
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