Seperation of variables / Alternative method to solve a DE

In summary, the conversation is about a problem in General Relativity involving computing Killing Vectors for a torus. The speaker is attempting to solve a differential equation and is looking for suggestions on how to successfully separate the equation or approach it differently. The expert provides the full solution, stating that F(\phi) = Const and g(\theta) = (C2*b*(a+b*cos(theta))-sin(theta)*C1)*(a+b*cos(theta))/b, where C1 and C2 are arbitrary constants. The speaker thanks the expert for their help.
  • #1
vhoffmann
2
0
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

[itex]
\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)
[/itex]

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

[itex]
\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)}
[/itex]

If anyone could suggest a way of successfully seperating this equation or a different approach to solving it, I'd be grateful.
 
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  • #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.
 
  • #3
Ah! The obvious choice of [itex]F(\phi) = Const[/itex] eluded me.

Thanks!
 

Related to Seperation of variables / Alternative method to solve a DE

1. How does separation of variables help in solving differential equations?

Separation of variables is a method used to solve differential equations by separating the variables on both sides of the equation. This helps in simplifying the equation and solving it in a step-by-step manner.

2. What are the steps involved in using separation of variables to solve a differential equation?

The steps involved are:

  1. Separating the variables on both sides of the equation
  2. Integrating each side of the equation separately
  3. Simplifying the equation and solving for the variables
  4. Substituting the values of the variables back into the original equation to check for accuracy

3. Are there any limitations to using separation of variables to solve differential equations?

Yes, separation of variables can only be used for certain types of differential equations that can be separated into distinct variables. It may not work for all types of differential equations.

4. Can you provide an example of using separation of variables to solve a differential equation?

Sure, let's take the differential equation dy/dx = 2x. We can separate the variables to get dy = 2x dx. Then, we can integrate both sides to get y = x^2 + C, where C is the constant of integration. This is the general solution to the differential equation.

5. Is there an alternative method to solve differential equations other than separation of variables?

Yes, there are other methods such as using substitution, using the method of undetermined coefficients, and using series solutions. The choice of method depends on the type of differential equation and the initial conditions given.

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