Final answer:
The LTI system characterized by the equation (D² + 4D + 4)y(t) = Df (t) has a characteristic polynomial of s² + 4s + 4, leading to a characteristic equation of s² + 4s + 4 = 0. The system has a repeated root at s = -2, resulting in characteristic modes e⁻²t and te⁻²t.
Step-by-step explanation:
The given system equation for a Linear Time Invariant (LTI) system is (D² + 4D + 4)y(t) = Df (t). To find the characteristics of the system, we begin by looking at the homogeneous equation, which is found by setting f(t) to zero.
The characteristic polynomial is obtained by equating the left-hand side of the homogeneous equation to zero and replacing D with the variable 's', which symbolizes the complex frequency domain representation in the Laplace Transform. Therefore, the characteristic polynomial is s² + 4s + 4.
Subsequently, the characteristic equation is the characteristic polynomial set to zero, hence: s² + 4s + 4 = 0.
The characteristic roots or eigenvalues are the solutions to the characteristic equation. For this system, we can factor the characteristic polynomial as (s + 2)², suggesting a repeated root at s = -2.
Considering this, the characteristic modes are the responses of the system based on the roots of the characteristic equation. Since we have a repeated root, the characteristic modes of the system are e⁻²t and te⁻²t which correspond to the system's natural response.
In conclusion, the characteristic polynomial is s² + 4s + 4, the characteristic equation is s² + 4s + 4 = 0, the characteristic roots are at s = -2 (with multiplicity two), and the characteristic modes are expressed by e⁻²t and te⁻²t.