In some online notes recently, I came across a nice demonstration of separation of variables in spherical coordinates to solve the Laplace's equation ($\nabla^2V = 0$). The general solution to this when there is azimuthal symmetry is of the form
$V(r,\theta) = \left(Ar^L + Br^{-(L+1)} \right)\left(P_{L}\cos\theta \right)$
where $P_{L}$ are the Legendre polynomials. They specific solution is then specified by the boundary conditions. My question is this - what if we had instead a constant inhomogeneous term, and the equation to be solved was instead
$\nabla^2 V = c$
where c is a constant. In this case, is it possible to use a similar approach to yield a solution, and how would one go about determining this? Alternatively, if it's not possible, why is the reason? Apologies in advance if this is a silly question, I am rather rusty on the rules around homogeneous and inhomogeneous solutions.
As pointed out in the comments, the usual way to solve an inhomogeneous solution is to use the Green's function for the differential operator, which depends on the region in which the differential equation is to be solved.
If the region has spherical symmetry, however, one can just look for a particular solution that has the same symmetry: i.e. a function of $r$. One then solves the equation $$ \nabla^2 v(r) = \frac{1}{r^2} (r^2v')' = c, $$ which has particular solution $v(r)=r^2/6$.
In either case, the homogeneous solution is then included to make the equation satisfy the boundary conditions, as one would with an ordinary differential equation. The boundary conditions can also be incorporated directly using the Green's function using Green's identities, see also this section of the Wikipedia article on Green's functions.