How Does the Hamiltonian Affect the Time Evolution of a Qubit's Density Matrix?

[whatisreality]

Homework Statement

In a computational basis, a qubit has density matrix
$\rho = \left( \begin{array}{ccc} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{2} \\ \end{array} \right)$
At t=0. Find the time dependence of $\rho$ when the Hamiltonian is given by $AI+BY$, $A$ and $B$ are constants, $Y$ is the Pauli matrix
$\left( \begin{array}{ccc} 0 & -i \\ i & 0 \\ \end{array} \right)$
Then the hint says to write the matrix in the form $\rho = \frac{1}{2} (I+a(t)X+b(t)Y+c(t)Z)$ where
$X= \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$,
$Z= \left( \begin{array}{ccc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)$
and derive differential equations for the coefficients.

Homework Equations

The Attempt at a Solution

I thought I should be using the time shift operator $U(t,t_0) = e^{-\frac{i}{\hbar}H(t-t_0)}$, and calculating $\rho(t) = U\rho(t_0) U^{\dagger}$, except I'm not exactly sure what the exponential of my particular Hamiltonian would mean. But if I write the density matrix in the form suggested by the hint then I don't know what to do with it from there to get a differential equation. Is it something to do with the Schrödinger equation? Should I sub it into that and treat it like a state $\psi$?

Thank you for any help, I really appreciate it, I'm a bit lost!

 

[eys_physics]

The exponential of the Hamiltonian (in this given in matrix form) is defined by its Taylor expansion. That is,
$exp(A)=\sum_{k=0}^\infty A^k/k!$ for a matrix $A$