Chapter 13.3, Problem 13E

To determine

The solution of the differential equation $r^2\frac{d^{2}R}{d{r}^2}+2r\frac{dR}{dr}+\left[k^2{r}^2-n\left(n+1\right)\right]R=0$ such that the three dimensional form of Helmholtz’s partial differential equation is ${\nabla }^{2}u+k^{2}u=0$ and $u\left(r,\theta ,\varphi \right)=R\left(r\right)\Theta \left(\theta \right)\Phi \left(\varphi \right)$.