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Math Help - ODE solution of exponential form

  1. #1
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    Question ODE solution of exponential form

    Hello everybody

    I'm trying to solve the following differential equation derived from a first order kinetic process:

    \tau {dm(t) \over dt} = m(\infty) - m(t)

    where m(\infty) is the steady-state of m(t), that I guess, for the moment could be treated as a constant.

    I know that the solution to the ODE is:

    m(t) = m(\infty) -\big(m(\infty)-m(0)\big ) \exp({-t/\tau} )

    being m(0) the initial condition.

    I've tried to solve the equation with a piece of paper, but get stuck in the process. Im principle, this equation should be easily solved by separation of variables...



    What I tried so far:
    First, separation of variables

     \frac{dm(t)}{m(\infty) - m(t)} = \frac{dt}{\tau}

    applying the integration limits:

     \int\limits_{0}^{t} \frac{dm(t)}{m(\infty) - m(t)}  =\int\limits_{0}^{t} \frac{dt}{\tau}

    solving the integral

    \log\frac{m(\infty) - m(t)}{m(\infty) - m(0)}  =\frac{t}{\tau}

    and applying exponential to both sides of the equation leads to:

    \frac{m(\infty) - m(t)}{m(\infty) - m(0)}  =\exp(t/\tau)

    organizing...
     m(\infty) - m(t) = \big(m(\infty) - m(0)\big) \exp(t/\tau)

    finally, solving for m(t)

     m(t) = m(\infty) - \big(m(\infty) - m(0)\big) \exp(t/\tau)

    which is almost the solution, but the exponential is raised to t/\tau and not to -t/\tau as it should be.

    I would appreciate if somebody would point out where is the mistake in the calculation

    Thanks

    Jose
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  2. #2
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    Re: ODE solution of exponential form

    Quote Originally Posted by jguzman View Post
    Hello everybody

    I'm trying to solve the following differential equation derived from a first order kinetic process:

    \tau {dm(t) \over dt} = m(\infty) - m(t)

    where m(\infty) is the steady-state of m(t), that I guess, for the moment could be treated as a constant.

    I know that the solution to the ODE is:

    m(t) = m(\infty) -\big(m(\infty)-m(0)\big ) \exp({-t/\tau} )

    being m(0) the initial condition.

    I've tried to solve the equation with a piece of paper, but get stuck in the process. Im principle, this equation should be easily solved by separation of variables...



    What I tried so far:
    First, separation of variables

     \frac{dm(t)}{m(\infty) - m(t)} = \frac{dt}{\tau}

    applying the integration limits:

     \int\limits_{0}^{t} \frac{dm(t)}{m(\infty) - m(t)}  =\int\limits_{0}^{t} \frac{dt}{\tau}

    solving the integral

    \log\frac{m(\infty) - m(t)}{m(\infty) - m(0)}  =\frac{t}{\tau}
    This integration is wrong. Letting " u= m(\infty)- m" gives du= -dm. You have the sign wrong in the integral so the fraction in the logarith m is inverted. You should have:
    \log\left(\dfrac{m(\infty)- m(0)}{m(\infty)- m(t)}\right)= \dfrac{t}{\tau}

    and applying exponential to both sides of the equation leads to:

    \frac{m(\infty) - m(t)}{m(\infty) - m(0)}  =\exp(t/\tau)

    organizing...
     m(\infty) - m(t) = \big(m(\infty) - m(0)\big) \exp(t/\tau)

    finally, solving for m(t)

     m(t) = m(\infty) - \big(m(\infty) - m(0)\big) \exp(t/\tau)

    which is almost the solution, but the exponential is raised to t/\tau and not to -t/\tau as it should be.

    I would appreciate if somebody would point out where is the mistake in the calculation

    Thanks

    Jose
    Thanks from jguzman
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  3. #3
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    Re: ODE solution of exponential form

    Fantastic! Thank you very much for your help!
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