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Math Help - How to find the function v=f(t,k) for which dv/dt = k/v

  1. #1
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    How to find the function v=f(t,k) for which dv/dt = k/v

    Hi,

    I'm looking for an expression where v is a function of t and k, and the derivative dv/dt = k/v.

    Any hints in how to solve this? Thank you.
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    Quote Originally Posted by fine1 View Post
    Hi,

    I'm looking for an expression where v is a function of t and k, and the derivative dv/dt = k/v.

    Any hints in how to solve this? Thank you.
    If k is merely a constant then the differential equation separates:
    vdv = kdt

    Simply integrate.

    -Dan
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  3. #3
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    Even if k is a function of t, you can still separate. Your solution will have an integral in it though...
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    Well, if f is a function of both k and t, then it should br \frac{\partial f}{\partial t}= \frac{k}{v} which can still be integrated- except that the "constant of integration" may be a function of k.
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    Thanks to all for your help In my case k is not a function of t. The solution is simply v = squareroot(2kt).
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by fine1 View Post
    Thanks to all for your help In my case k is not a function of t. The solution is simply v = squareroot(2kt).
    Actually that's only one possibitlity. You have left out the arbitrary constant...
    v dv = k dt

    (1/2)v^2 = kt + C

    v = sqrt{2kt + 2C} = sqrt{2kt + A}
    where A is arbitrary.

    -Dan
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    Quote Originally Posted by topsquark View Post
    Actually that's only one possibitlity. You have left out the arbitrary constant...
    v dv = k dt

    (1/2)v^2 = kt + C

    v = sqrt{2kt + 2C} = sqrt{2kt + A}
    where A is arbitrary.

    -Dan
    Quite so. Thanks
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