Final answer:
The given system is linear because it satisfies the properties of additivity and homogeneity.
Step-by-step explanation:
The given system is not linear. To determine if a system is linear, we need to check if it satisfies the properties of additivity and homogeneity. Additivity means that if we apply two different inputs to the system, the output will be the sum of the outputs when each input is applied separately. Homogeneity means that if we scale the input by a constant factor, the output will be scaled by the same factor.
Let's check if the system satisfies additivity. If we apply two different inputs x1(t) and x2(t), the output y(t) will be:
y(t) = u(t) ∫ 0 to t e^-(t-a) x1(a+1) da + u(t) ∫ 0 to t e^-(t-a) x2(a+1) da
Using the linearity property of integration, this can be rewritten as:
y(t) = u(t) ∫ 0 to t e^-(t-a) (x1(a+1) + x2(a+1)) da
Since the output is the sum of the outputs when each input is applied separately, the system satisfies additivity. Now let's check if it satisfies homogeneity. If we scale the input by a constant factor, the output will be:
y(t) = u(t) ∫ 0 to t e^-(t-a) (k*x(a+1)) da
Using the linearity property of integration, this can be rewritten as:
y(t) = k*u(t) ∫ 0 to t e^-(t-a) x(a+1) da
Since the output is scaled by the same factor as the input, the system satisfies homogeneity. Therefore, the given system is linear. The answer is a) 1.