Well, I don't understand the electrostatics, but here is another physical heuristic:
Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Formerly I had written "charge density",
but now I am not sure if that's right.]
I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of temperature in the interior.
This satisfies a diffusion (or heat) equation, but in words:
"After a time \delta, the new temperature is obtained by averaging the old temperature along a circle of radius \sqrt{\delta}."
This process converges under reasonable conditions, as time goes to infinity, to the solution of the Dirichlet problem. Anyway, we are led to the Brownian-motion proof of the existence, which I personally find rather satisfying. Another personal comment: I think one should always take "physical heuristics" rather seriously.
[In response to Q.Y.'s comments below, which were responses to previous confused remarks that I made: neither the electric field nor the Columb potential is a multiple of the charge density on the boundary: the former is a vector, and in either
case imagine the charge on the boundary to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]
This is a second degree linear ODE with rational function coefficients, and this problem has been completely solved algorithmically by Kovacic in the mid-eighties:
Kovacic, Jerald J. "An algorithm for solving second order linear homogeneous differential equations." Journal of Symbolic Computation 2, no. 1 (1986): 3-43.
All the major computer algebra systems implement this, and (not surprisingly) Mathematica does not find a solution in closed form in the generality you give. However, since your $c_i$ are ``known real numbers'', perhaps you are lucky, and a closed for solution is known in your special case. The relevant Mathematica incantataion is "DSolve", as in:
DSolve[y''[x] + (c1/x^2 + c2/(1 - x)^2 + c3 (1/x + 1/(1 - x))) y'[
x] + (c4 (1/x + 1/(1 - x)) + c5/x^2 + c6/(1 - x)^2) y[x] ==
0, y[x], x]
Best Answer
If the ODE is linear --and the notion of «explicit» refers to Liouvillian solutions (towers of iterated quadrature and exponential of meromorphic functions)-- then its differential Galois group (Picard-Vessiot theory) must be a solvable algebraic subgroup of $GL_n (\mathbb C)$. Such subgroups are rare: they define a proper algebraic subvariety. The defining equations correspond to trivial commutations relations, e.g $[G,G]=0$ in the Abelian case, encoding the tower of normal subgroups intervening in the definition of solvable group.
In the non-linear context the intuition is the same: nice «transverse structures» are rare.
Edit: for a statement regarding the non-linear case, see my paper http://fr.arxiv.org/abs/1308.6371v2, Corollary C.