Question

For a continuous-time linear time-invariant (LTI) system with input x(t) and ouipat y(f) govemed by the differential equation

d²/dt²y(t)+7d/dty(t)+12y(t)=x(t)
for t≥0.
What is the transfer function?

Answer 1

The transfer function of the given LTI system with the differential equation d²/dt²y(t) + 7d/dty(t) + 12y(t) = x(t) is H(s) = 1/(s² + 7s + 12).

Step-by-step explanation:

The transfer function of a continuous-time linear time-invariant (LTI) system that is governed by the differential equation d²/dt²y(t) + 7d/dty(t) + 12y(t) = x(t) for t ≥ 0 can be found using the Laplace transform.

Applying the Laplace transform to both sides of the equation and assuming initial conditions are zero, we get s²Y(s) + 7sY(s) + 12Y(s) = X(s), where Y(s) is the Laplace transform of y(t) and X(s) is the Laplace transform of x(t). The transfer function H(s) is then defined as the ratio of the output Y(s) to the input X(s), yielding H(s) = Y(s)/X(s) = 1/(s² + 7s + 12).