Results 1 to 2 of 2

Thread: Differential Model

  1. #1
    Junior Member
    Aug 2009

    Differential Model

    A classical problem in the calculus of variations is to find the shape of a curve such that a bead, under the influence of gravity, will slide from point A(0,0) to point B (x1,y1) in the least time. It can be shown that a nonlinear differential for the shape y(x) of the path is $\displaystyle y[1 + (dy/dx)^2] = k$, where k is a constant. First solve for dx in terms of y and dy and then use the substitution $\displaystyle y = ksin^2(theta)$ to obtain a parametric form of the solution. The curve turns out to be a cycloid.

    Not entirely sure what to do with this problem, mainly b/c I don't really understand what the problem is asking, which includes what a cycloid is...or how you find the parametric solution for one.

    Not sure if this is right, but it's what I've done so far:
    $\displaystyle (dy/dx)^2 = k/y - 1$
    $\displaystyle dy/dx = (k/y - 1)^1^/^2$
    $\displaystyle dx = dy/[(k/y - 1)^1^/^2]$

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Jun 2010
    CT, USA
    Looking good so far. I'm assuming that the problem defines y positive downward, so that you want the positive square root for the derivative. If y is positive up, then you'd need to take the negative square root, since the bead obviously has to be going down under the influence of gravity. So you have

    $\displaystyle dx=\dfrac{dy}{\sqrt{k/y-1}}=dy\sqrt{\dfrac{y}{k-y}}.$

    The suggested substitution is $\displaystyle y=k\sin^{2}(\theta),$ with

    $\displaystyle dy=2k\sin(\theta)\cos(\theta)\,d\theta.$ We obtain the new DE

    $\displaystyle dx=2k\sin(\theta)\cos(\theta)\,d\theta\sqrt{\dfrac {k\sin^{2}(\theta)}{k-k\sin^{2}(\theta)}}.$

    Can you continue from here?

    Here is a link to information about cycloids.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Model with First Order Differential Equation
    Posted in the Differential Equations Forum
    Replies: 7
    Last Post: May 31st 2011, 06:18 PM
  2. Non-linear process model with Linear measurement model
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Sep 1st 2009, 11:32 PM
  3. Replies: 3
    Last Post: Apr 15th 2009, 01:18 AM
  4. differential equations- single compartment model
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Mar 12th 2009, 07:52 AM
  5. Replies: 4
    Last Post: Apr 29th 2008, 08:08 PM

Search Tags

/mathhelpforum @mathhelpforum