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Math Help - Rewriting a ODE in -t

  1. #1
    Senior Member Dinkydoe's Avatar
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    Rewriting a ODE in -t

    It feels like a stupid question...

    given the system:
    \frac{dx}{dt} = y
    \frac{dy}{dt} = Ay-x(1-x)

    with x=x(t), y=y(t)

    How would I rewrite the system in, say \hat{t}=-t?
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  2. #2
    A Plied Mathematician
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    Well, you have an autonomous system there, meaning that you don't have to worry about the RHS's so much. Try this:

    \displaystyle{\frac{dy}{d\hat{t}}=\frac{dy}{dt}\,\  frac{dt}{d\hat{t}}.}
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  3. #3
    Senior Member Dinkydoe's Avatar
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    Doesn't this just give... \frac{dy}{d\hat{t}} = -\frac{dy}{dt}

    Don't we also want to have y(\hat{t}) in our equations?

    How do we deal with that?
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  4. #4
    A Plied Mathematician
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    Can't you just make the substitution outright? I'm thinking

    \displaystyle{-\frac{dy}{d\hat{t}}=y(\hat{t})}

    \displaystyle{-\frac{dy}{d\hat{t}} = Ay(\hat{t})-x(\hat{t})(1-x(\hat{t}))}.

    Wouldn't that do the job?
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  5. #5
    Senior Member Dinkydoe's Avatar
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    I don't see how we could just make that step...

    \frac{dy}{d\hat{t}}=-\frac{dy}{dt},\frac{dx}{d\hat{t}}=-\frac{dx}{dt}

    Gives...

    -\frac{dx}{d\hat{t}} = y(t)
    -\frac{dy}{d\hat{t}}= Ay(t)-x(t)(1-x(t))

    I don't see how we could just subtitute t= \hat{t} in the eqations...
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  6. #6
    A Plied Mathematician
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    Do x and y represent physical quantities of any sort?
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  7. #7
    Senior Member Dinkydoe's Avatar
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    No, they do not specifically represent anything...

    So, if we want to rewrite a system in \hat{t}, i don't see why y(t) can simply be replaced by y(\hat{t})
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  8. #8
    A Plied Mathematician
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    No, they do not specifically represent anything...
    A pity, or we could invoke this.

    I don't have any other answer, I'm afraid, other than I think you can just do it. I don't have a justification. This is the physicist part of me talking, I freely admit.
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