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Analytical Solution to ODEs

  1. Apr 23, 2007 #1
    I have a couple ODEs that I need to solve. I was probably just going to put them into mathematica, but I like finding the analytical way also. The first one is

    [tex]\frac{d}{dx}\left( \frac{(y + \lambda)y'}{\sqrt{1+y'^2}} \right) = \sqrt{1+y'^2}[/tex]

    Lambda is a constant and y' is dy/dx. I suppose that after all the quotients and products are evaluated it could be separated, but that is a lot of work if there is a nice trick to employ.

    The other one, the one I am actually curious about since it is nonlinear, is actually a pair of ODEs

    [tex]ma^2(sin^2\theta \ddot{\phi}+ 2\phi sin\theta cos\theta) = 0[/tex]
    and
    [tex]ma^2 \ddot{\theta} = -mga sin\theta + 2a^2 sin\theta cos\theta \dot{\phi}^2[/tex]

    In this problem, m, g, and a are constants. I need to solve the coupled equations for both phi and theta. I was thinking that maybe I could convert the equations into first order ODEs, and then solve the system of equations, but I'm not sure how to deal with the nonlinearity of the [tex] \dot{\phi}^2[/tex] and trig functions.

    These came from assigned physics HW problems (Legrangians/Hamiltonians), so I assume they can be solved without mathematica.

    *Nevermind about the first one, I solved it with separation and integration tables. The solution was [tex] y = c cosh((x-b)/c) - \lambda[/tex] in case anyone guessed.
     
    Last edited: Apr 24, 2007
  2. jcsd
  3. Apr 24, 2007 #2

    quasar987

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    I had to do a similar problem earlier this semester and it was implicit in the question that we could do a small angle approximation.

    Also, you might want to check out arildno's complicated answer (post #7) about how we can drop the squares angular velocity also in the small angle aprox:

    https://www.physicsforums.com/showthread.php?t=152090
     
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