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Math Help - differential equation help,

  1. #1
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    differential equation help,

     \frac{dS}{dt} = \frac{K_{cat}E_{0}S}{K_{m} + S}

     \frac{dP}{dt} = \frac{K_{cat}E_{0}S}{K_{m}+S}

    with intial conditions  S(0) = S_{0}  P(0) = p_{0)

    show that S and P satisfy a conservation law given by S+P = c,

    for some constant c, and find the value of c. Hence determine  \lim_{t\  \infty}  P(t)

    I know that if you add the two equations you get a 0, is that what c equals?
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  2. #2
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    Re: differential equation help,

    From \frac{dS}{dt}= \frac{K_{cat}E_0S}{K_mS} and \frac{dP}{dt}= \frac{K_{cat}E_0S}{K_mS} subtracting, \frac{dS}{dt}+ \frac{dP}{dt}= 0.
    We also have S+ P= c for constant c, differentiating both sides, we have \frac{dS}{dt}- \frac{dP}{dt}= 0.

    So subtracting the second equation from the first, we have 2\frac{dP}{dt}= 0 from which it follows that P(t) is a constant.
    Thanks from Tweety
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  3. #3
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    Re: differential equation help,

    Sorry I have written out the equation wrong, I have now corrected it, also I dont understand how you got a minus, after differentiating both sides s+p = c.?
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  4. #4
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    Re: differential equation help,

    Quote Originally Posted by Tweety View Post
     \frac{dS}{dt} =- \frac{K_{cat}E_{0}S}{K_{m} + S}

     \frac{dP}{dt} = \frac{K_{cat}E_{0}S}{K_{m}+S}

    with intial conditions  S(0) = S_{0}  P(0) = p_{0)

    show that S and P satisfy a conservation law given by S+P = c,

    for some constant c, and find the value of c. Hence determine  \lim_{t\  \infty}  P(t)

    I know that if you add the two equations you get a 0, is that what c equals?
    The first equation should have a minus infront of it
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