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Math Help - Nonlinear ODE from optimization problem

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    Nonlinear ODE from optimization problem

    I'm working through the Brachistochrone Problem but I get stuck at the differential equation:

    (1+y'^2)y=K^2 (eq. 13)

    The solution is supposed to be in the form of two parametric functions x(\theta) and y(\theta), so I tried switching to polar coordinates and manipulating the equation in various ways. But that lead me to some hairy integrals, and I want to find out if I'm on the right track before I continue.

    What's the best approach here?
    Last edited by TwoPlusTwo; September 17th 2013 at 12:34 PM.
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    Re: Nonlinear ODE from optimization problem

    Quote Originally Posted by TwoPlusTwo View Post
    I'm working through the Brachistochrone Problem but I get stuck at the differential equation:

    (1+y'^2)y=K^2 (eq. 13)

    The solution is supposed to be in the form of two parametric functions x(\theta) and y(\theta), so I tried switching to polar coordinates and manipulating the equation in various ways. But that lead me to some hairy integrals, and I want to find out if I'm on the right track before I continue.

    What's the best approach here?
    Hint: The equation is separable.

    -Dan
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    Re: Nonlinear ODE from optimization problem

    Quote Originally Posted by topsquark View Post
    Hint: The equation is separable.

    -Dan
    Thanks. So after separating I get:

    x=\int_0^y\sqrt{\frac{y}{K^2-y}}dy

    But this feels like another dead end. Even if I could solve the integral, I'm not sure I would know what to do with the result...
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    Re: Nonlinear ODE from optimization problem

    Look at the article that you posted earlier. Take a look at equations 30 - 33. Specifically take a look at the substitution in equation 31. That's the key to integrating the problem, but I'm afraid that I can't make it work. If you like you can work backward and verify that x( \theta) \text{ and } y( \theta ) satisfy the differential equation. (It's fussy about the details, but isn't too hard overall.) That's as far as I can go with it.

    -Dan
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    Re: Nonlinear ODE from optimization problem

    Quote Originally Posted by topsquark View Post
    Look at the article that you posted earlier. Take a look at equations 30 - 33. Specifically take a look at the substitution in equation 31. That's the key to integrating the problem, but I'm afraid that I can't make it work. If you like you can work backward and verify that x( \theta) \text{ and } y( \theta ) satisfy the differential equation. (It's fussy about the details, but isn't too hard overall.) That's as far as I can go with it.

    -Dan
    Thanks, I will give it a try. It could be that when people first solved this, they just guessed at a solution that made sense (a cycloid), and then checked to see if it worked.
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