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Seperation of variables / Alternative method to solve a DE

  1. Dec 4, 2009 #1

    This question is in the context of General Relativity problem. I'm attemping to compute the Killing Vectors for a Torus. After some juggling around I ended up with the following differential equation

    \frac{d}{d \theta} \left( \frac{ (a+b \cos \theta) \sin \theta }{b} F(\phi) + g(\theta) \right) + \frac{d}{d\phi} f(\phi) = 2 \left( \frac{-b \sin \theta }{ a + b \cos \theta } \right) \left( \frac{ (a + b \cos \theta) \sin \theta }{b } F(\phi) + g(\theta) \right)

    where [itex]g(\theta)[/itex] and [itex]f(\phi)[/itex] are what I'm after. Note that [itex]F(\phi)[/itex] is the primitive of [itex]f(\phi)[/itex] (i.e., a second order equation).

    I suspect the equation is seperable, so I've been attempting to rewrite the equation accordingly, but haven't made much headway.

    Farthest I got was

    \left( \frac{a}{b} \cos \theta + 1 \right) F(\phi) + \frac{d}{d\phi} f(\phi) = - \frac{d}{d\theta} g(\theta) - 2 \frac{b \sin \theta}{ a + b \cos \theta } g(\theta)}

    If anyone could suggest a way of successfully seperating this equation or a different approach to solving it, I'd be grateful.
  2. jcsd
  3. Dec 4, 2009 #2
    The full solution of your DE is as follows

    F(phi)=C1 , that is, f(phi)=0 ,

    g(theta) = (C2*b*(a+b*cos(theta))-sin(theta)*C1)*(a+b*cos(theta))/b ,

    where C1 and C2 are arbitrary constants.
  4. Dec 14, 2009 #3
    Ah! The obvious choice of [itex]F(\phi) = Const[/itex] eluded me.

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