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Differential Equations Help

  1. Sep 28, 2005 #1
    I have the following to solve:

    [tex]\frac{dx}{dt}=-\alpha xy;\quad y=y_0e^{-\beta t};\quad x(0)=x_0[/tex]

    I separate variables and come up with:

    [tex]\frac{dx}{x}=-\alpha y_0e^{-\beta t}dt[/tex]

    [tex]\ln{x}=-\alpha y_0\int e^{-\beta t}dt=\frac{\alpha y_0}{\beta}e^{-\beta t}+C[/tex]

    ...so for a final answer I come up with:

    [tex]x=x_0\exp{\left(\frac{\alpha y_0}{\beta}e^{-\beta t}\right)}[/tex]

    ..however the book says that the answer is:

    [tex]x=x_0\exp{\left(\frac{-\alpha y_0\left(1-e^{-\beta t}\right)}{\beta}\right)}[/tex]

    I cannot find where I went wrong, any ideas?

    Thanks a lot.
  2. jcsd
  3. Sep 28, 2005 #2


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    You didn't evaluate the time integral at both limits.
  4. Sep 28, 2005 #3


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    Good catch. I stared at it for a few minutes and couldn't figure it out.
  5. Sep 28, 2005 #4
    ...both limits?
  6. Sep 28, 2005 #5


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    You have:

    [tex]x(t)=K\text{Exp}\left[\frac{\alpha y_0}{\beta}e^{-\beta t}\right][/tex]



    Now, carefully substitute that initial value into the equation to solve for K.
  7. Sep 28, 2005 #6
    Ahh, I must have put the x0 term there prematurely. Thanks for the help everyone, I have it now. :smile:
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