91Ó°ÊÓ

The Zero-Input Response, often referenced as the natural response, indicates how systems react exclusively due to the initial conditions, with no external force or input being applied after the initial moment. For instance, consider the differential equation: \( y' (t) - 3 y (t) = t^2 u (t) \); and initial condition \( y(0) = 2 \). To find the zero-input response, we set the right-hand side to zero, solving the homogeneous equation: \[ y' (t) - 3y(t) = 0. \] The characteristic equation for this is: \[ r - 3 = 0 \] which yields \( r = 3 \). Therefore, a particular solution to the homogeneous equation is: \[ y_{\text{hom}}(t) = C e^{3t} \] with the initial condition \( y(0) = 2 \); we obtain: \[ y_{\text{hom}}(0) = C e^{0} = C = 2 \]. Hence, the zero-input response is: \[ y_{\text{hom}}(t) = 2 e^{3t}. \]

The Zero-State Response describes how the system reacts to an external input with all initial conditions set to zero. Let's revisit our example: \( y' (t) - 3 y (t) = t^2 u (t) \); assuming \( y(0)=2 \). To discover the zero-state response, we approach it through the inhomogeneous part while assuming zero initial conditions. We can guess a particular solution: \[ y_p (t) = At^2 + Bt + C. \] Substituting \( y_p \) into the differential equation and solving for coefficients A, B, and C yields: \[ (2At + B) - 3(At^2 + Bt + C) = t^2. \] Equating coefficients provides: \[ A = -\frac{1}{3}, B = 0, C =0 \]. Thus, the zero-state response comes as: \[ y_{\text{zs}} (t) = -\frac{1}{3} t^2. \]

A system is time-invariant if its behavior and characteristics do not change over time. To verify if a system is time-invariant, check if shifting the input results in a correspondingly shifted output. Mathematically, shifting an input \( u (t) \) to \( u(t - t_0) \) should lead to a solution \( y(t - t_0) \). For example, if our original differential equation \( y'(t) - 3y(t) = t^2 u(t) \) is given a delayed input \( u(t - t_0) \), the corresponding output should also be shifted by \( t_0 \).

A homogeneous differential equation has all terms involving the dependent variable or its derivatives. It tends to describe the system's behavior intrinsic to its initial state, without any external inputs. As an illustration, in our case: \[ y' (t) - 3 y(t) = 0 \] is a homogeneous equation. The characteristic equation is derived to find solutions: \[ r - 3 = 0 \]; leading to \( r = 3 \). The general solution then is: \( y_{\text{hom}}(t) = Ce^{3t} \). Homogeneous equations are crucial as they aid in assessing the zero-input response.

The Particular Solution pertains to solving the non-homogeneous differential equation. It specifically accounts for how the system dynamically reacts to an external input. Generally, we guess the form of the particular solution, substitute it into the differential equation, and solve for constants. In our example: \[ y' (t) - 3 y (t) = t^2 u(t). \] A guessed particular solution form: \[ y_p (t) = At^2 + Bt + C \]; substituting and equating coefficients for \( A, B, \) and \( C \) yields the particular solution: \[ y_p (t) = -\frac{1}{3} t^2. \] Summing this with the homogeneous solution gives the complete response of the system, reflecting both the zero-input and zero-state responses.