Solve for n: Finding Value of n for Equation V

[utkarshakash]

Homework Statement

Find the value of n so that the equation V=r^n(3 \cos ^3 \theta -1) satisfies the relation
$$\dfrac{\partial}{\partial r} \left( r^2 \dfrac{\partial V}{\partial r} \right) + \dfrac{1}{\sin \theta}\dfrac{\partial}{\partial \theta} \left( \sin \theta \dfrac{\partial V}{\partial \theta} \right)=0$$

Homework Equations

The Attempt at a Solution

The final equation after differentiation comes out to be
$$n(n+1)(3 \cos ^3 \theta -1) = 18 \cos \theta (2\cos ^2 \theta -1) $$

What should I substitute for θ ?

 

[Simon Bridge]

Why should you need to substitute something for theta?
Recheck your algebra... you final relation is saying there is no value of n that will make V satisfy the PDE.

 

[utkarshakash]

Why should you need to substitute something for theta?
Recheck your algebra... you final relation is saying there is no value of n that will make V satisfy the PDE.

I've already checked it thrice but couldn't find any error! Can you please show me where I'm going wrong?

 

[Ray Vickson]

I've already checked it thrice but couldn't find any error! Can you please show me where I'm going wrong?

How can anybody tell where you went wrong? You have not shown your work in detail.

I (or Maple) get something completely different from your result.