I am currently using Signals and Systems by Alan Oppenheim as a reference to learn about LTI systems.
Before introducing systems represented by linear constant coefficient differential equations, the author first considers general systems of the form $x(t) \rightarrow y(t)$. When discussing linearity of systems, he introduces the "zero in-zero out" property of linear systems (not necessarily time invariant or causal), that is, given a linear system, if the input $x(t)$ is zero for all $t$, then the output $y(t)$ must be zero for all $t$. In addition, a linear system is causal if and only if $y(t) = 0$ for all $t < t_0$ if $x(t) = 0$ for all $t < t_0$, a condition known as initial rest, from which I concluded that initial rest for linear systems implies causality but not time invariance.
Now given a linear constant coefficient differential equation $$Ly(t) = x(t),$$ where $L$ is a linear differential operator with constant coefficients, the author emphasizes that although the differential equation is linear, the system that it represents may or may not be linear depending on the initial conditions. I find this confusing because given two input output pairs $x_1(t) \rightarrow y_1(t)$ and $x_2(t) \rightarrow y_2(t)$, we have $L(ay_1(t) + by_2(t)) = ax_1(t) + bx_2(t)$ for any constants $a, b \in \mathbb C$ by virtue of the linearity of the operator $L$. It seems that $ax_1(t) + bx_2(t) \rightarrow ay_1(t) + by_2(t)$ regardless of the initial conditions, why is the system not necessarily linear in that case? What am I missing here?
In addition, the author states that for the system to be linear, the initial conditions have to be zero, and for it to be causal and time invariant in addition to being linear, its initial conditions must obey initial rest. But isn't initial rest in general a condition required for causality, not time invariance? Besides, doesn't this imply that a differential equation can only represent a causal system if the input is zero before some time? Given an input that is nonzero for all time, how do we impose causality via initial conditions that correspond to initial rest? I hope my question makes sense. Thank you in advance.