Let's have the following equation: $$ u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = f(r), $$ where $r$ is polar radius.
Method of Green's function leads to $$ u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = \delta(r). \qquad (1) $$ Here I have little trouble.
It seems that the solution is $C_{1}I_{0}(\alpha r) + C_{2}K_{0}(\alpha r)$, where $I_{0}, K_{0}$ are Infeld and Macdonald functions respectively. But I don't know how to choose constants. It's obvious that Macdonald functions is correct solution of $(1)$, but I don't sure that I can set $C_{2},C_{1}$ to $1, 0$.
$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\dd}{{\rm d}}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}% \newcommand{\yy}{\Longleftrightarrow}$ $\ds{u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = f}$ $$ u\pars{r} = {\rm u_{p}}\pars{r} + \int_{-\infty}^{\infty}{\rm G}\pars{r,r'}f\pars{r'}\,\dd r' $$ where ${\rm u}_{p}\pars{r}$ is a solution of $\ds{u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u = 0}$ which satisfies the given boundary conditions. ${\rm G}\pars{r,r'}$ is the Green function which satisfies $$ \pars{\partiald[2]{}{r} + {1 \over r}\partiald{}{r} - \alpha^{2}}G\pars{r,r'} = \delta\pars{r - r'} $$ ${\rm G}\pars{r,r'}$, as a function of $r$, satisfies homogeneous boundary conditions.