Final answer:
The transfer function of the system, Y(s)/X(s), is given by (s³ + 4s² + 6s + 8) / (s³ + 3s² + 5s + 1).
Step-by-step explanation:
The provided differential equation represents a linear time-invariant system. To determine the transfer function Y(s)/X(s), we perform the Laplace transform on both sides of the differential equation, assuming zero initial conditions. Each term on the left side undergoes the Laplace transform, as does each term on the right side.
The Laplace transform of the left side results in the polynomial (s³ + 3s² + 5s + 1), while the Laplace transform of the right side yields (s³ + 4s² + 6s + 8). By dividing the Laplace transform of the output (Y(s)) by the Laplace transform of the input (X(s)), we obtain the transfer function Y(s)/X(s), expressed as (s³ + 4s² + 6s + 8) / (s³ + 3s² + 5s + 1).
In summary, the transfer function characterizes the frequency domain behavior of the system. The expression (s³ + 4s² + 6s + 8) / (s³ + 3s² + 5s + 1) signifies how the system responds to a given input, offering valuable insights for system analysis and control design without the need for latex formatting.