For system of differential equation as follows:\begin{align} \frac{\partial}{\partial t} \begin{pmatrix}\rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{00}\end{pmatrix} &= -\tau i \begin{pmatrix} \rho_{10} - \rho_{01} & \rho_{11} - \rho_{00} \\ \rho_{00} - \rho_{11} & \rho_{01} -\rho_{10} \end{pmatrix} + \tau^2 \begin{pmatrix} \rho_{11} & -\frac{1}{2}\rho_{01} \\ -\frac{1}{2}\rho_{10} & -\rho_{11} \end{pmatrix} \end{align}
Alternative form:
$\rho_{00}'(t)=-\tau i (\rho_{10}(t)-\rho_{01}(t))+\tau^2 \rho_{11}(t)$
$\rho_{01}'(t)=-\tau i (\rho_{11}(t)-\rho_{00}(t))-\tau^2 \frac{1}{2}\rho_{01}(t)$
$\rho_{10}'(t)=-\tau i (\rho_{00}(t)-\rho_{11}(t))-\tau^2 \frac{1}{2}\rho_{10}(t)$
$\rho_{11}'(t)=-\tau i (\rho_{01}(t)-\rho_{10}(t))-\tau^2 \rho_{11}(t)$
Question: How to obtain the analytic solutions ? I tried but failed, it seems the solution is large ? In Mathematica, it gives the solution extremly large with many terms, is it reasonable ?