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 $g(r)$?



ODE with Bessel decay
Maple 2018 solves the ODE under consideration:



```
dsolve(((D@@2)(w))(r)+(D(w))(r)/r-k^2*w(r) = -g(r))

```

$$w \left( r \right) ={{\ I}\_{0}\left(kr\right)}{\it \\_C2}+{{ K}\_{0
}\left(kr\right)}{\it \\_C1}-\int \!{{ K}\_{0}\left(kr\right)}g
 \left( r \right) r\,{\rm d}r{{ I}\_{0}\left(kr\right)}+$$ $$\int \!{
{ I}\_{0}\left(kr\right)}g \left( r \right) r\,{\rm d}r{{ K}\_{0}
\left(kr\right)}
 $$


The assumption $w(r)$ decays exponentially fast implies the constant ${\it \\_C2} $ equals zero because of



```
asympt(BesselI(0, k*r), r, 2)

```

$$\frac 1 2\,{\frac {\sqrt {2}\sqrt {{k}^{-1}}{{\rm e}^{kr}}\sqrt {{r}^{-1}}}{
\sqrt {\pi}}}+O \left( \left( {r}^{-1} \right) ^{3/2} \right)
 $$


and



```
asympt(BesselK(0, k*r), r, 1)

```

$$\frac 1 2\,{\frac {\sqrt {2}\sqrt {\pi}{{\rm e}^{-kr}}\sqrt {{r}^{-1}}}{
\sqrt {k}}}+O \left( \left( {r}^{-1} \right) ^{3/2} \right)
 $$


Hope that would be useful.


Addition. The Mathematica 11.3 command



```
AsymptoticDSolveValue[ D[w[r], {r, 2}] +D[w[r], r]/r - k^2*w[r] == -Exp[-k*r]/Sqrt[r],w[r], 
{r, Infinity, 1}, Assumptions -> k > 0]

```

disproves the suggestion of the question. See the long output here <https://www.dropbox.com/s/on0jz2kpb7bin7l/AS.pdf?dl=0> and pay your attention to the term $\frac{\sqrt{r}}{2 k} $.


