# Guaranteeing uniqueness of solution of ODE.

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?

• try to google for a "existence and uniqueness theorem for Delay differential equation". I'm not an expert in DDE, so I'm not sure if there is such a theorem for a general case. For some cases it is definitely exists, e.g. google.co.il/… – Michael Medvinsky Nov 26 '15 at 21:28