1. The problem statement, all variables and given/known data
Find the value of n so that the equation [itex]V=r^n(3 \cos ^3 \theta -1)[/itex] satisfies the relation
$$\dfrac{\partial}{\partial r} \left( r^2 \dfrac{\partial V}{\partial r} \right) + \dfrac{1}{\sin \theta}\dfrac{\partial}{\partial \theta} \left( \sin \theta \dfrac{\partial V}{\partial \theta} \right)=0$$
2. Relevant equations
3. 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 ?
Find the value of n so that the equation [itex]V=r^n(3 \cos ^3 \theta -1)[/itex] satisfies the relation
$$\dfrac{\partial}{\partial r} \left( r^2 \dfrac{\partial V}{\partial r} \right) + \dfrac{1}{\sin \theta}\dfrac{\partial}{\partial \theta} \left( \sin \theta \dfrac{\partial V}{\partial \theta} \right)=0$$
2. Relevant equations
3. 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 ?
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