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Homework Help: Differential Equations and growth constant

  1. Feb 14, 2010 #1
    dS/dt = kS - W

    How do I solve this problem if k is the growth constant and W is also constant?

    Here is what I have so far, but I dont think its quite right:

    dS/dt = kS - W

    (kS - W)-1dS = 1dt

    1/k*ln(kS - W) = t + C

    ln(kS - W) = kt + C1

    kS - W = C2ekt

    S = (C2ekt + W) / k

    S = C3ekt + W/k

    I think I did the math right but when I try plugging a few numbers in for the constant, the derivative doesnt match the function. Am I suppose to use the integrating factor for this problem?
  2. jcsd
  3. Feb 14, 2010 #2


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    But it does! Your solution satisfies the equation dS/dt = kS - W. Try plugging S in to that equation; you'll see that both sides are equal.
  4. Feb 14, 2010 #3
    Plug S into what equation?

    I just think its wrong because if I take the derivative of S = C3ekt + W/k, then I would get S' = kekt, which isnt S' = kS - W. The W isnt suppose to disappear.
  5. Feb 14, 2010 #4


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    The term kS contributes a +W which cancels with the -W, leaving just the exponential.
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