Suppose that system has the input/output relation as follows
$$\sum_{k=0}^{N}a_k \frac{d^ky(t)}{dt^k} = \sum_{k=0}^{M}b_k \frac{d^kx(t)}{dt^k}$$
Where $a_k, b_k \in \mathbb{R}$.
Obviously we need additional conditions to determine completely the input-output relationship for the system and those auxiliary conditions specify whether the system is LTI and causal or not.
The definition of initial rest condition for a system in general is that if $x(t) = 0$ for $t\lt t_0$ then $y(t) = 0$ for $t\lt t_0$.
First of all is it true that the initial rest condition for the all systems implies $y(t_0) = 0$? Also I have seen expressions like $y({t_0}^+)$ and $y({t_0}^-)$ but I don't know how they relate to $y(t_0)$. Another question is that whether the initial rest condition is necessary and sufficient for the mentioned system to be causal and LTI? What's the rigorous proof, then? I searched on the Internet but didn't found a proof or a clear explanation.