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Trouble with Solving a Differential Equation

  1. Feb 5, 2006 #1
    I'm having trouble with the following textbook problem:

    The only basic approach for solving differential equations that I know is to "separate and integrate", that is, have one side with P's and a dP, and the other side with t's and a dt (constants anywhere throughout), and then integrate both sides. So, what I was hoping to get was kM * something * dt = kP * something * dP. However, it looks like that's not going to happen. Could someone help me solve this thing?

    I know that if "M is the maximum level of performance", then logically [tex]\lim_{t \rightarrow \infty} P(t) = M[/tex]. However, I'm still not sure how to actually find the function P(t).

    Thanks for any help you can give. :smile:
    Last edited: Feb 5, 2006
  2. jcsd
  3. Feb 5, 2006 #2

    It's easily separable just divide by by M-P and you have a pretty standard integral to find the function P.
  4. Feb 5, 2006 #3
    Ah, of course! I guess my first mistake was distributing the k. Dividing brings me to [tex]\frac{dP}{M - P} = kdt[/tex], and integrating both sides gives me [tex]-ln|M - P| = kt + c[/tex]. Going from there:

    [tex]ln|M - P| = -kt - c[/tex]
    [tex]e^{-kt - c} = M - P[/tex]
    [tex]P = M - e^{-kt - c}[/tex]
    I think I'm doing something very wrong here. The answer in the back of the book is [tex]P(t) = M - Me^{-kt}[/tex].

    I'm thinking I've made a really stupid mistake, but I can't find it. Can anyone help?
    Last edited: Feb 5, 2006
  5. Feb 6, 2006 #4


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    Have you left out part of the problem- like, say, the information that the subject knew nothing to start with? That is, that P(0)= 0.

    [tex]P(t) = M - e^{-kt - c}= M- (e^{-c})e^{-kt}[/tex]
    and we can write that as
    [tex]P(t)= M- C e^{-kt}[/tex]
    where [itex] C= e^{-c}[/itex].

    Then, if P(0)= 0,
    [tex]P(0)= M- Ce^0= M- C= 0[/tex]
    so C= M.

    Actually that's not relevant to the final question: what is the limit of P(t) as t goes to infinity? That answer does not depend on C.
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