Final answer:
The given systems are linear based on the properties of superposition and homogeneity.
Step-by-step explanation:
The given systems are:
y₁(t) = x(t)x(t - 2)y₂(t) = t²x(t)
To determine whether these systems are linear or nonlinear, we need to check if they satisfy the properties of linearity.
A system is linear if it follows the properties of superposition and homogeneity.
Property 1: Superposition
To test for superposition, we check if the system satisfies the equation:
y(t) = a₁y₁(t) + a₂y₂(t)
where a₁ and a₂ are constants.
Let's substitute the given systems into the equation:
y(t) = a₁(x(t)x(t - 2)) + a₂(t²x(t))
Expanding and simplifying:
y(t) = a₁x(t)x(t - 2) + a₂t²x(t)
Since the equation satisfies superposition, the system is linear based on the first property.
Property 2: Homogeneity
To test for homogeneity, we check if the system satisfies the equation:
y(at) = ax(t)
Let's substitute the given systems into the equation:
y(at) = a(x(at))x(at - 2) = a(x(t)a(t - 2)) = ax(t)x(at - 2)
The equation satisfies homogeneity, so the system is linear based on the second property.
Therefore, the given systems are linear based on the properties of superposition and homogeneity.