Some are actually boundary conditions. When you impose them the system becomes overdetermined. So instead solve using just the ICs, then impose the BCs to find out what can be determined about the parameters.

m[x_] := (L w) x(*up F*)- (w x) (x/2)
ode = {e i y''[x] == m[x] + M, y[0] == 0, y'[0] == 0}

soln = First[DSolve[ode, y[x], x]]
 (* Out[978]= {y[x] -> (12 M x^2 + 4 L w x^3 - w x^4)/(24 e i)} *)

Now for the BCs:

paramEqns = {D[y[x] /. soln, x] /. x -> L, 
  y[x] /. soln /. x -> 2*L, D[y[x] /. soln, x] /. x -> 2*L}

(* Out[976]= {(24 L M + 8 L^3 w)/(24 e i), (48 L^2 M + 16 L^4 w)/(
 24 e i), (48 L M + 16 L^3 w)/(24 e i)} *)

Solve[paramEqns == 0]

(* Out[977]= {{L -> 0}, {M -> -((L^2 w)/3)}} *)

Discarding the first as non-physical (or at least not of interest), we now have a necessary condition involving {M,L,w}.