How can you solve $r^2v^{\prime\prime} + (n-1)rv^\prime + r^2v=0$?

Question

Let $n\ge 2$ be an integer. I am trying to solve the ODE $$ r^2v^{\prime\prime}(r) + (n-1)rv^\prime(r) + r^2v(r)=0. $$ This arises when attempting to find a fundamental solution for $\Delta + c$ where $c>0$. I know that the solution involves Bessel functions, but I am having a hard time finding an appropriate transform. Any advice or references on how to proceed would be appreciated.

Answer 1

Let $v(r)=r^{a} y(r)$. Taking the derivatives and then substituting into your equation, we see that choosing $a=1-\frac{n}{2}$ leads to Bessel's equation for $y(r)$. The index of the solutions $Y_\alpha(r)$ and $J_\alpha(r)$ ends up being given by $\alpha^2=(1-\frac{n}{2})^2$.