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I was wondering about the nature of stable systems (in the BIBO sense) that are causal with memory for which we wish to represent them by LCCDE (if they may exist). How frequent do LCCDE exist such that the derivative of the output is related to a time-shifted version of the output and are they resolved numerically?

I wish to further note that in saying that the system has memory, I don't mean it in the sense that only the input is shifted, but also I am looking for an output that is already shifted. To provide an example : $$ y'(t)+K_{1}y(t-\alpha)+K_{2}x(t-\beta)=0 $$ where $\alpha<\beta$ and $K_{1},K_{2}\in\mathbb{R}$ non-zero constants

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