Suppose we are given the Hamiltonian
$$H=f \frac{\text{Tr}\sigma_x \rho}{\text{Tr}\rho}\sigma_x,$$
where $\rho$ is the density matrix, and $\sigma_x$ is the Pauli matrix $$ \sigma_x= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, $$
and $f$ is a coupling constant. The time evolution of the density matrix is given by
$$\frac{d \rho}{dt}=if[\rho,\sigma_x]\frac{\text{Tr}\sigma_x \rho}{\text{Tr}\rho}.$$
How do I proceed from here? How do I integrate this equation?
Your Hamiltonian does not make sense: The Hamiltonian cannot depend on the density matrix, but is a linear operator acting on the Hilbert space (the same space on which the density operator is acting). Similarly, the von Neumann equation is a linear differential equation for $\rho$ (just as all of quantum mechanics is linear), which your equation isn't.
Otherwise, a correct von Neumann equation is a linear differential equation, and thus its integral is an exponential function.
