Radial eigenfunctions of the Laplace operator in spherical coordinates

Question

I am trying to find solutions for the following ODE (which was derived trying to find the fundamental solution of the PDE $\Delta u+cu=0$ where $c > 0$ see below for the approach)

The ODE is

$v''(r)+\frac{2}{r} v'(r)+cv(r)=0$

I am not really sure how to solve this. I haven't done a computational ODE course, and I did look around but couldn't find any resource which tells a way to approach such problems clearly. Any help or at least a hint or a resource would be appreciated.

As for solving the PDE, I was looking for radial solutions and thats how I arrived at the ODE. (Mainly following the Laplace Equation method from Evans). Please let me know if there is anything wrong with this approach too

Thank You

Answer 1

Let us set $w(r) = r v(r)$. Using the product rule, we have the relationships $$ w'(r) = v(r) + r v'(r) $$ and $$ w''(r) = 2 v'(r) + r v''(r) . $$ Thus, the differential equation rewrites as a linear equation with constant coefficients $$ w''(r) + c w(r) = 0 \, , $$ whose solutions are $w(r) = A\cos(r\sqrt{c}) + B\sin(r\sqrt{c})$. Division by $r$ then provides $v(r)$.

Answer 2

I'm no expert, so I'm likely to miss something. But the usual thing would be to assume that $v(r) $ is analytic, substitute the series into your equation balance the indices, and get a recursion for the coefficients of $v$.

Answer 3

Write it in the original form $$-r^{-2} \partial_r(r^2\partial_r v(r)) = k^2\ v(r) $$ Its a trivial equation for $r^2 v(r)$ $$ \partial_r(r^2\partial_r v(r)) = k^2 (r^2\ v(r)) $$

Works generally for the radial operator in n dimensions with $$-\Delta_r =- r^{-n+1} \partial_r (r^{n-1}\ \partial_r v(r)) = k^2\ v(r) $$