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Math Help - Brachistochrone ODE

  1. #1
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    Brachistochrone ODE

    Hey guys,
    I'm a bit rusty on differential equations and today I was helping a friend solving ODE when we stumbled on this one:

    (1+ (y')^2)y = k^2

    The problem asked to solve for y'. Can you point me in the right direction? Thanks!
    Last edited by TomAlso; September 29th 2011 at 08:08 PM.
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  2. #2
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    Re: Brachistochrone ODE

    suppose y'=p we get
    (1+p^2)y=k^2 gives y=\frac{k^2}{1+p^2}
    now differentiating wrt x and putting \frac{dy}{dx}=psolve the equation in p and x
    Last edited by Ackbeet; September 30th 2011 at 06:37 AM. Reason: Retitled.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Brachistochrone ODE

    Quote Originally Posted by TomAlso View Post
    Hey guys,
    I'm a bit rusty on differential equations and today I was helping a friend solving ODE when we stumbled on this one:

    (1+ (y')^2)y = k^2

    The problem asked to solve for y'. Can you point me in the right direction? Thanks!
    With simple steps You obtain the pair of ODE...

    y^{'} = \pm \sqrt{\frac{k^{2}}{y}-1} (1)

    ... and in each of them the variables can be separated...

    Kind regards

    \chi \sigma
    Last edited by Ackbeet; September 30th 2011 at 06:37 AM. Reason: Retitled.
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