I tried searching for a solution to this type of problem online but was unsuccessful. I almost found a solution here, but it requires the coefficient matrix (the square one in the equation below) to have distinct eigenvalues which doesnt hold for this problem. Anyway here's the problem:
$$ \begin{bmatrix} \partial \rho/\partial t \\[0.3em] \partial V/\partial t \end{bmatrix} + \begin{bmatrix} V & \rho \\[0.3em] 0 & V\end{bmatrix} \begin{bmatrix} \partial \rho/\partial r \\[0.3em] \partial V/\partial r \end{bmatrix}=\begin{bmatrix} -2\rho V/r \\[0.3em] A\rho r \end{bmatrix} $$
Here $A$ is a constant. Any help or trick to modify the mehod in the link to suit this problem would be appreciated.