This is very similar to what you provided.
g = 10;
system = Table[D[y[x, t], {t, 2}] == -100 (2*y[x, t] - y[x - 1/13, t] - y[x+1/13, t]) -
10 (2*D[y[x, t], {t, 1}] - D[y[x - 1/13, t], {t, 1}] - D[y[x+1/13, t], {t, 1}]) + g,
{x, -1/2 + 1/13, 1/2 - 1/13, 1/13}];
functions = Table[y[x, t], {x, -1/2 + 1/13, 1/2 - 1/13, 1/13}];
DSolve[system, functions, t]
It doesn't incorporate any initial or boundary conditions and thus should only return the general solution and it appears to be very slow because of the number of equations involved, but it doesn't appear to give any warning or error messages.
Perhaps if it finishes you might be able to then apply your conditions and obtain the specific solution that you are looking for. Or you might try adding conditions one at a time and try to interpret the warning and error messages to see if you can make those additions acceptable.
