The initial rest assumption corresponds to a zero-valued auxiliary condition being imposed at a time determined in accordance with the input signal. In this problem we show that if the auxiliary condition used is nonzero or if it is always applied at a fixed time (regardless of the input signal) the corresponding system cannot be LTI. Consider a system whose input x (t) and output y (t) satisfy the first-order differential equation (P2.33-1).
(a) Given the auxiliary condition y(1) = 1, use a counterexample to show that the system is not linear.
(b) Given the auxiliary condition y(1) = 1, use a counterexample to show that the system is not time invariant.
(c) Given the auxiliary condition y(1) = 1, show that the system is incrementally linear.
(d) Given the auxiliary condition y(1) = 0, show that the system is linear but not time invariant.
(e) Given the auxiliary condition y(0) + y(4) = 0, show that the system is linear but not time invariant.
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