DSolve[{b1*x[t] + a*(Derivative[1][x][t] - Derivative[1][y][t]) + Derivative[2][x][t] == c1*E^(I*omega*t), b2*y[t] + a*(-Derivative[1][x][t] + Derivative[1][y][t]) + Derivative[2][y][t] == c2*E^(I*omega*t)},{x[t], y[t]},t]