How to use following equation by using Green's function? 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 $1, 0$. 
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$\ds{u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = f}$
$$
u\pars{r}
=
{\rm u_{p}}\pars{r} + \int_{-\infty}^{\infty}{\rm G}\pars{r,r'}f\pars{r'}\,\dd r'
$$
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.
