Can we say that a Linear Constant Coefficient Difference Equation can always represent a Linear Shift Invarient system ? Are there any conditions which need to be satisfied additionally by these kind of equations to be able to do that?
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WHat about final rest condition? (i.e if x(t) = 0, for t > t0 then y(t) is also 0 for t > t0 ) whether now the differential equation satisfies LSI or not?
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It is not necessary that a linear constant coefficient difference equation must represent an LTI system. It will represent an LTI system if and only if the solution satisfies the initial rest condition, namely if $x[n] = 0$ for $n<n_0$, then $y[n] = 0$ for $n<n_0$