Let us consider a pair of $n$-th order linear homogeneous ODEs with variable coefficients:
$$y^{(n)}(t) = a_1(t) y(t) + a_2(t) y'(t) + ... + a_n(t) y^{(n-1)}(t)$$
$$y^{(n)}(t) = b_1(t) y(t) + b_2(t) y'(t) + ... + b_n(t) y^{(n-1)}(t)$$
where $y^{(n)}(t)$ denotes $n$-fold differentiation w.r.t $t$.
We may assume that $t \in [0,1]$, and that the coefficients are all smooth in $t$.
Are there general criteria for determining, given $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$, whether the two ODEs have any common solution $y(t)$?