For $\displaystyle \nabla^2 u(r,\theta, \phi)=0,\;u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi) $

But I have issue with this, for spherical coordinates:

$\displaystyle \nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)$

Let $\displaystyle u=R(r)Y(\theta,\phi) $ where $\displaystyle Y(\theta,\phi) $ is the spherical harmonics.

$\displaystyle \Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0$

and

$\displaystyle \frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partia l {\phi}^2}+\mu Y=0$

For Euler equation: $\displaystyle r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-\mu R=0$ where $\displaystyle \mu=n^2$. and the solution is $\displaystyle R=r^n$.

Here, because of the condition, only ##\mu=n(n+1)## is used for bounded solution.

$\displaystyle r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+\;r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R$

Which gives

$\displaystyle R=r^{(n+1/2)}$

What have I done wrong?