For a continuous-time system described by y'' + 3 y' + 2 y = x, which is its transfer function H(s)?

• A. 1 / (s^2 + 3s + 2)
• B. 1 / (s^2 - 3s + 2)
• C. (s^2 + 3s + 2)
• D. s^2 + 3s + 2

Answer: (A)

When a continuous-time linear system is described by a differential equation, the transfer function is found from the input-output relationship in the Laplace domain under zero initial conditions. For y'' + 3y' + 2y = x, take the Laplace transform with zero initial conditions: s^2 Y(s) + 3 s Y(s) + 2 Y(s) = X(s). Factor the output term: Y(s) [s^2 + 3s + 2] = X(s). The transfer function is the ratio Y(s)/X(s), which gives H(s) = 1 / (s^2 + 3s + 2). This can also be seen as the inverse of the differential operator, reflecting how the input is transformed to yield the output. The other forms would imply a different relationship between Y(s) and X(s) and hence correspond to different equations, so they don’t match the given system. The denominator factors as (s+1)(s+2), so the poles are at -1 and -2, indicating a stable system.