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ODE - Brachistochrone Problem

  1. May 5, 2010 #1
    y[1+(y')^2] = k

    First solve for dx in terms of y and dy, an then use the substitution y = ksin2(θ) to obtain a parametric form of the solution. The curve turns out to be a cycloid.

    My attempt:

    (y')^2 = k/y-1

    dy/dx = sqrt(k/y-1)

    dx = dy/[sqrt(k/y-1)]

    then substitute y = ksin^2(θ)

    dx = dy/[sqrt(1/sin^2(θ)-1]

    dx = dy/[sqrt(cos^2(θ)/sin^2(θ)]

    dx = dy/cot(θ)

    I don't know where to go from here, but the parametric form of y should equal k(1-cos(θ)).
    Any help would be appreciated.
  2. jcsd
  3. May 5, 2010 #2


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    Homework Helper

    shouldn't you sub in for dy in terms of dθ
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