Vtyjkyuk

I have the following three differential equations for three functions $f_0(r)$, $f_1(r)$, and $f_2(r)$:

$$
beginalign*
-kappa M delta^(3)(textbfr)
&=
-m^2 f_1
+ frac2 m f_1r
- frac2 f_1r^2
-frac2 partial_r f_1r
- frac6 m f_2r
+ frac2 f_2r^2
- 4 m partial_r f_2
+ frac6 partial_r f_2r
+ 2 partial_r^2 f_2,
\
0
&=
- m^2 f_0
- frac2 m f_0r
+ frac2 partial_r f_0r
+ frac2 f_1r^2
+ 2 m^2 f_2
+ frac2 m f_2r
- frac2 f_2r^2
- frac2 partial_r f_2r,
\
0
&=
-m f_0
+ partial_r f_0
- 2 m r partial_r f_0
+ r partial_r^2 f_0
- m f_1
+ m^2 r f_1
+ partial_r f_1
\
&quad
+ 2 m f_2
- 2 partial_r f_2
+ 2 m r partial_r f_2
- r partial_r^2 f_2,
endalign*
$$

where $kappa$, $m$, and $M$ are positive constants and $r geq 0$. The only boundary condition is that the functions must vanish in the limit $r to infty$. The details of the differential equations themselves aren't important for my question.

To handle the Dirac delta function in the first equation, I know to solve the homogeneous version of the equation first, and then integrate to fix the constant. Expanding the three functions in a power series and solving for the coefficients yields

$$
beginalign*
f_0(r) &= fracBr,\
f_1(r) &= -B
left(frac1m r^2 + frac1m^2 r^3right),\
f_2(r) &= fracB2
left(frac1r + frac1m r^2 + frac1m^2 r^3right),
endalign*
$$

where $B$ is an arbitrary constant.

Now I need to fix the constant $B$ by integrating the first equation. Plugging these functions into the first differential equation yields

$$
beginalign*
-kappa M delta^(3)(textbfr)
&=
B
left(
-frac2mr^2
- frac3r^3
- frac2m r^4
+ frac3m^2 r^5
- 2m partial_rleft(frac1rright)
+ frac3r partial_rleft(frac1rright)
right.
\&quad
left.
- 2 partial_rleft(frac1r^2right)
+ frac5m r partial_rleft(frac1r^2right)
- frac2m partial_rleft(frac1r^3right)
+ frac5m^2 r partial_rleft(frac1r^3right)
right.
\&quad
left.
+ partial_r^2left(frac1rright)
+ frac1m partial_r^2left(frac1r^2right)
+ frac1m^2 partial_r^2left(frac1r^3right)
right).
endalign*
$$

If I actually evaluate the above derivatives, the right hand side of the equation vanishes, which is to be expected. However, I don't see how to actually do the integration on both sides to fix the constant.
I know the correct answer is $B = dfrac14pidfrac23 kappa M$.

How am I to get this answer? I figured the identity

$$
nabla^2 frac1r = -4pi delta^(3)(textbfr).
$$

might be useful, but I can't seem to get it to work.