Given the $n$th order ODE:
$$\sum _{0\leq i \leq n}a_ix^{(i)}=0,\quad x(t_1) = \xi_1, x^{(1)}(t_2)=\xi_2,\ldots,x^{(n-1)}(t_{n-1})=\xi_{n-1} $$
I know that if $t_1=\cdots=t_{n-1}$, then uniqueness of solutions is guaranteed.
But what if $t_i\neq t_j\forall j\neq i $. Is there a way to find bounds on the differences of the $t_i$'s to guarantee uniqueness?