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Math Help - [SOLVED] Differential Equation Help

  1. #1
    Member Maccaman's Avatar
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    [SOLVED] Differential Equation Help

    Hi all,
    Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

    Here it is:
    "find the equation of R(t) when

    \frac{dE}{dt}= -\gamma_{b}E where  E(0) = E_0 and  \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p ,

    \frac{dS}{dt}= -\gamma_{s}S where  S(0) = S_0 and  \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p

    and

     \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS
    where  M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p are all constants.

    Now I know that

     S(t) = S_0 e^{-\gamma_{s} t}

     E(t) = E_0 e^{-\gamma_{b} t}

    and then by substituting these values into  \frac{dR}{dt} we obtain

     \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}

    this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
    Last edited by Maccaman; September 20th 2008 at 05:48 PM. Reason: Latex Error
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  2. #2
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    Quote Originally Posted by Maccaman View Post
    Hi all,
    Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

    Here it is:
    "find the equation of R(t) when

    \frac{dE}{dt}= -\gamma_{b}E where  E(0) = E_0 and  \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p ,

    \frac{dS}{dt}= -\gamma_{s}S where  S(0) = S_0 and  \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p

    and

     \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS
    where  M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p are all constants.

    Now I know that

     S(t) = S_0 e^{-\gamma_{s} t}

     E(t) = E_0 e^{-\gamma_{b} t}

    and then by substituting these values into  \frac{dR}{dt} we obtain

     \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}

    this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
    Just integrate with respect to t.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Maccaman View Post
    Hi all,
    Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

    Here it is:
    "find the equation of R(t) when

    \frac{dE}{dt}= -\gamma_{b}E where  E(0) = E_0 and  \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p ,

    \frac{dS}{dt}= -\gamma_{s}S where  S(0) = S_0 and  \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p

    and

     \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS
    where  M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p are all constants.

    Now I know that

     S(t) = S_0 e^{-\gamma_{s} t}

     E(t) = E_0 e^{-\gamma_{b} t}

    and then by substituting these values into  \frac{dR}{dt} we obtain

     \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}

    this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
    Well, as the right hand side doesn't depend on R all you need to do is integrate both sides with respect to t.

    -Dan
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    Member Maccaman's Avatar
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    ahhhhhh, of course I feel so stupid now
    thanks
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Maccaman View Post
    ahhhhhh, of course I feel so stupid now
    thanks
    There is nothing more difficult than seeing the obvious after you have done something complicated.

    -Dan
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  6. #6
    Member Maccaman's Avatar
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    Is this answer correct? (my integration rules are a bit dodgy)


     R(t) = - \frac{\gamma_{b_1} E_0 e^{-\gamma_b t}}{\gamma_b} - \frac{\gamma_{s_1} M S_0 e^{-\gamma_s t}}{\gamma_s}
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Maccaman View Post
    Is this answer correct? (my integration rules are a bit dodgy)


     R(t) = - \frac{\gamma_{b_1} E_0 e^{-\gamma_b t}}{\gamma_b} - \frac{\gamma_{s_1} M S_0 e^{-\gamma_s t}}{\gamma_s}
    Looks good to me.

    -Dan
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