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First order differential problem question

  1. Aug 27, 2009 #1
    question :

    As the salt KNO3 dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation dx/dt = 0.8x - 0.004x^2

    if x=50 when t=0, how long will it take for an additional 50g of salt to dissolve.

    ok, here I'm encountering a problem in the differential part, I don't know how to solve the differential question.. so far, what I have done...

    using separable differential equations..

    1/(0.8x - 0.004 x^2) dx = dt

    and integral both sides...

    1/(0.8x - 0.004 x^2) dx = t+c

    I don't know to integrate the right side, I use the integral calculator and the answer is

    { 5 ln x - 5 ln (x-200) } / 4 so, the whole equation will be

    5 ln x - 5 ln (x-200) = 4 (t+c)

    but, I think it doesn't work when I substitute x=50 when t=0. cause the ln (x-200) can't be minus..

    so, please advise me

  2. jcsd
  3. Aug 27, 2009 #2
    When you perform an integral to get a ln function, it is always ln of the absolute value of the variables. So your equation should read 5 ln |x| - 5 ln |x-200} = 4 (t+c).
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