If $X(t)$ is an $n \times n$ matrix solving linear homogeneous ODE $$ \frac{d}{dt} X(t) = A(t)X(t), $$ then for $\det X(t)$ we have Liouville's formula: $$ \frac{d}{dt} \det X(t) = \text{tr} A(t) \det X(t). $$
The determinant $\det X(t)$ is the constant term of the characteristic polynomial $\det(X(t)-\lambda I_n)$. I want to know differential equations for other coefficients of this polynomial $a_i(t) = \text{tr} \wedge^i X(t)$. Is something like this known?
Update: Even in the simplest case $n=2$ it is not clear hot to find ODE for $\text{tr} X(t)$. It is clear that $\frac{d}{dt} \text{tr} X(t)=\text{tr}(A(t)X(t))$, but it is not clear how to proceed from this.