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Math Help - simplification of ODE

  1. #1
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    simplification of ODE

    I saw this in a book:
     \frac{dy}{dt} = -\frac{y}{\tau} +f(t),
    can be simplified to y = f(t)\tau, if one is interested only in the time scale slower than \tau,

    I don't know how this is got, could someone help me with it? Many thanks!
    Last edited by wclayman; March 13th 2010 at 12:45 AM.
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  2. #2
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    Quote Originally Posted by wclayman View Post
    I saw this in a book:
     \frac{dy}{dt} = -\frac{y}{\tau} +f(t),
    can be simplified to y = f(t)\tau, if one is interested only in the time scale slower than \tau,

    I don't know how this is got, could someone help me with it? Many thanks!
    Presumably it means nothing more than that the change of y with respect to t is very small and in the approximation stated means equal to 0. So you have:

    \frac{dy}{dt}=0=-\frac{y}{\tau}+f(t)

    From which the relation [imath]y=f(t)\cdot \tau[/imath]

    Coomast
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  3. #3
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    Thanks! I get it.
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  4. #4
    MHF Contributor chisigma's Avatar
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    The DE can be written as...

    y^{'} + \frac{y}{\tau} = f(t) (1)

    ... whose solution is...

     y(t) = y (0) \cdot e^{-\frac{t}{\tau}} + \varphi (t) (2)

    ... where...

    \varphi (t) = \int_{0}^{t} e^{-\frac{t-u}{\tau}}\cdot f(u)\cdot du (3)

    If t << \tau the exponential terms in (2) and (3) can be approximated at 1 so that is...

    y(t) \approx y(0) + \int_{0}^{t} f(u)\cdot du (4)

    Kind regards

    \chi \sigma
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  5. #5
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    Quote Originally Posted by chisigma View Post
    The DE can be written as...

    y^{'} + \frac{y}{\tau} = f(t) (1)

    ... whose solution is...

     y(t) = y (0) \cdot e^{-\frac{t}{\tau}} + \varphi (t) (2)

    ... where...

    \varphi (t) = \int_{0}^{t} e^{-\frac{t-u}{\tau}}\cdot f(u)\cdot du (3)

    If t << \tau the exponential terms in (2) and (3) can be approximated at 1 so that is...

    y(t) \approx y(0) + \int_{0}^{t} f(u)\cdot du (4)

    Kind regards

    \chi \sigma
    I am confused now, first, the assumption is actually  t \gg \tau , and following this way, I can't arrive at the above-mentioned conclusion.
    Last edited by wclayman; March 13th 2010 at 06:04 AM.
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  6. #6
    MHF Contributor chisigma's Avatar
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    What is [for me] difficult to understand is why Coomast sets in His post \frac{dy}{dt}=0... the same to say that y(t) is a constant ...

    Kind regards

    \chi \sigma
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  7. #7
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    I'm not sure now, ...
    The author of that book is coming to visit the day after tomorrow, I'll take that chance and consult him directly. After that I will post the what he says to me.
    I strongly suspect he had some typo or something! That solution is wrong.
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