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\[ \rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(t_1)]}\]  One can solve by iteratively repeatedly  plugging it the above equation  into itself: \[ \rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(0)] + \left(\frac{-i}{\hbar}\right)^2 \int_{0}^{t}{dt_1 \int_{0}^{t_1}{dt_2[H(t_1),[H(t_2),\rho(t_2)]]}}}\]