I need to verify that a function is a solution of a logistics DE

  1. 1. The problem statement, all variables and given/known data

    Verify by direct cauculation that if k, C, and d are constants, then the function P(t) = C/(1+d*e[tex]^{-kCt}[/tex]) is a solution of the logistic DE P' = kP(C-P).


    2. Relevant equations

    I don't think there are any for this problem. :)


    3. The attempt at a solution

    Okay, so ... uh ... I guess in this problem I should just be looking for the derivative of the original equation. So here goes ....

    P(t) = C/(1+d*e[tex]^{-kCt}[/tex])
    P(t) = C(1+d*e[tex]^{-kCt}[/tex])[tex]^{-1}[/tex] -- [I just moved the bottom part to the top.]
    P(t) = -(e[tex]^{-t}[/tex])[tex]^{-2}[/tex]*-1 (chain rule) <-- I think this is where I go wrong. C, k, and d are constants so I just made their derivaties one. Is that the right thing to do? Because somehow I get the feeling that the third line of work here isn't going to get me to the answer.
     
  2. jcsd
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