This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.

I would like to estimate the asymptotic behaviour of the solution of the following ODE $$ w''(r)+\frac{1}{r}w'(r)-k^{2}w(r)=-g(r)$$ where $g$ is a smooth positive function satisfying $$ g(r) = O\left(\frac{e^{-kr}}{\sqrt{r}}\right) \ \ \ \ \ r\to +\infty . $$

Assuming that $w(r)$ decays exponentially fast, is it true that w(r) has exactly the same asymptotic decay as g(r)?

Edit: The answers below suggest this is not true. Let me change a little bit the assumptions on $g(r)$: suppose that there exists $0<\alpha<1$ such that $$ g(r) = O\left(\frac{e^{-kr}}{r^{\alpha}\sqrt{r}}\right) \ \ \ \ \ r\to +\infty . $$ Can we now say that $w(r)=O\left(\frac{e^{-kr}}{\sqrt{r}}\right)$ ?