Final answer:
The given system is linear, time-invariant, and causal. The linearity is verified by checking the principle of superposition, while time invariance is confirmed by comparing time-shifted input and output signals. Causality is determined by observing that the output only depends on the current and past behavior of the input signals.
Step-by-step explanation:
This system is described by the differential equation:
d³y(t)/dt³ +t² dy(t)/dt+9y(t)=dx(t)/dt−8x(t)
Let's analyze the linearity, time invariance, and causality of this system:
Linearity:
To check for linearity, we need to verify if the differential equation satisfies the principle of superposition. This means that if we let y₁(t) and y₂(t) be two solutions to the equation, then c₁y₁(t) + c₂y₂(t) should also be a solution, where c₁ and c₂ are constants. Let's assume y₁(t) and y₂(t) are two solutions to the equation:
(D³ + t²D + 9)y₁(t) = (Dx(t) − 8x(t))
(D³ + t²D + 9)y₂(t) = (Dx(t) − 8x(t))
Now, let's consider the sum of c₁y₁(t) + c₂y₂(t):
(D³ + t²D + 9)(c₁y₁(t) + c₂y₂(t)) = c₁(D³ + t²D + 9)y₁(t) + c₂(D³ + t²D + 9)y₂(t) = c₁(Dx(t) − 8x(t)) + c₂(Dx(t) − 8x(t)) = (c₁ + c₂)(Dx(t) − 8x(t))
Hence, the differential equation satisfies the principle of superposition, and the system is linear.
Time Invariance:
A system is time-invariant if shifting the input signal by a time delay results in an equivalent shift in the output signal. To check for time invariance, we assume y(t) is a solution to the equation and consider a time-shifted input x(t - T):
(D³ + t²D + 9)y(t) = (Dx(t) − 8x(t))
(D³ + t²D + 9)y(t - T) = (Dx(t - T) − 8x(t - T))
Comparing the two equations, we can see that the time shift in the input signal results in an equivalent time shift in the output signal. Therefore, the system is time-invariant.
Causality:
A system is causal if its output only depends on the input signal for the current and past times. In the given system, the output y(t) depends on the derivative terms dy(t)/dt and dx(t)/dt. These derivative terms represent the current and past behavior of the input signals x(t) and y(t). Therefore, the system is causal.