Seperation of variables / Alternative method to solve a DE

vhoffmann

Hej,

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

[latex]
\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)
[/latex]

where [latex]g(\theta)[/latex] and [latex]f(\phi)[/latex] are what I'm after. Note that [latex]F(\phi)[/latex] is the primitive of [latex]f(\phi)[/latex] (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

[latex]
\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)}
[/latex]

If anyone could suggest a way of successfully seperating this equation or a different approach to solving it, I'd be grateful.

kosovtsov

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.

vhoffmann

Ah! The obvious choice of [latex]F(\phi) = Const[/latex] eluded me.

Thanks!