General solution of Poisson's equation How to find general solution of Poisson's equation in electrostatics.
$$
\nabla^2V=-\frac{\rho}{\epsilon_0}
$$
Where,
V  = electric potential
ρ  = charge density around any point
εₒ = absolute permittivity of free space.
 A: Poison's equation is a partial differential equation (PDE), so one solves it using the techniques of differential calculus (or something fancier). So, typically, one firstly solves the homogeneous part of the ODE in question and then secondly the non-homogeneous part, and then you combine the two solutions for a general solution. Of course, a charge density, $\rho(x,y,z)$, must be specified in order to solve the non-homogeneous part of the PDE. This is a routine exercise in introductory E&M, meaning there is a lot of information available about it:
Here's an explicit solution of Laplace's equation using the method of Separation of Variables http://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx
Here's a solution of Poisson's equation using the method of Green's Functions
http://farside.ph.utexas.edu/teaching/em/lectures/node31.html
I recommend Griffith's Intro to E&M textbook for further reference into how to solve specific boundary conditions for Poisson's equation, here's a pdf I found online (I hope this is okay to do) http://kestrel.nmt.edu/~mce/griffiths_4.pdf
For further reference into the theory of differential equations, see M. Boas' Mathematical Methods in the Physical Sciences https://www.amazon.com/Mathematical-Methods-Physical-Sciences-Mary/dp/0471198269
A decent intro to Differential Equations book is by R. Haberman https://www.amazon.com/Applied-Differential-Equations-Boundary-Problems/dp/032179706X
Edit: using the Green's function approach is usually taught/learned after the Separation of Variables approach. Green's functions require more familiarity with mathematical abstraction than does Separation of Variables. Plus, Separation of Variables allows the student to see the "nuts and bolts" of the ODE without getting lost in uniqueness and reciprocity theorems (especially when considering non-trivial charge distributions). The ardent student should learn the method of Green's functions, however, if they want to pursue graduate school (look in John David Jackson's Classical Electrodynamics).
A: See for Poisson's equation
http://farside.ph.utexas.edu/teaching/em/lectures/node31.html
The general solution for Poisson's equation is
$$ \phi (r) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho (r')}{|r-r'|}  d^3 r'    $$
Intuitively, decompose $\rho$ into point charges. Then each point charge gets 'smeared out' by 1/r to yield its potential. And finally the individual potentials from the point charges are summed together.
