Return to Answer

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} $.

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} $.

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.

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} $.

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 constants ${\it \_C1}$ and ${\it \_C2} $ equal 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.

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.