In the previous problem we saw that application of an auxiliary condition at a fixed tim...

In the previous problem we saw that application of an auxiliary condition at a fixed time (regardless of the input signal) leads to the corresponding system being not time-invariant. In this problem, we explore the effect of fixed auxiliary conditions on the causality of a system. Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation (P2.33-1). Assume that the auxiliary condition associated with the differential equation is y(0) = 0. Determine the output of the system for each of the following two inputs:

Observe that if y₁ (t) is the output for input x₁ (t) and y₂(t) is the output for input x₂(t), then y₁ (t) and y₂(t) are not identical for t < —1, even though x₁ (t) and x₂(t) are identical for t < —1. Use this observation as the basis of an argument to conclude that the given system is not causal.