Final answer:
System (a) is non-linear but time-invariant; (b) is both non-linear and time-variant; (c) is non-linear and time-variant; and (d) is linear and time-invariant.
Step-by-step explanation:
To determine if a system is linear and/or time-invariant, we evaluate the system using the definitions of linearity, which involves the properties of superposition (additivity and homogeneity), and time invariance, which means the system's output does not change when its input is shifted in time.
a. y(t) = 3x(t) + 1: This system is not linear because of the constant term '+1', which breaks the property of homogeneity. It is time-invariant because changing the time t to t+τ does not change the form of the output.
b. y(t) = 3 sin(t) x(t): This system is non-linear due to the multiplication by the time-varying term sin(t), and also it is time-variant because the sin(t) term introduces dependence on the specific time.
c. dy/dt + t y(t) = x(t): This differential equation represents a time-variant system because of the 't' multiplying the y(t) term, which causes the system to change with time. It is non-linear due to the presence of the term involving the product of 't' and 'y(t)'.
d. dy/dt + 2y(t) = 3 dx/dt: This is a linear differential equation because it only involves linear operations on y(t) and x(t). It is also time-invariant as there are no coefficients that change with time.