Question

For each of the systems specified by the following transfer functions, find the differential equation relating the output y(t) to the input x(t), assuming that the systems are controllable and observable:

H(s)= s+ 5 / s² +3s+8

Answer 1

The differential equation relating y(t) to x(t) for the system with the transfer function H(s) = (s + 5) / (s² + 3s + 8) is y''(t) + 3y'(t) + 8y(t) = x'(t) + 5x(t).

Step-by-step explanation:

To derive the differential equation relating the output y(t) to the input x(t) for the system with the transfer function H(s) = (s + 5) / (s² + 3s + 8), we must utilize the Laplace transform properties. Given a transfer function H(s), the differential equation can be obtained by taking the inverse Laplace transform and setting up the relationship in the time domain.

The transfer function represents the Laplace transform of the output over the Laplace transform of the input, H(s) = Y(s) / X(s). To find the differential equation, we multiply both sides by X(s) and then take the inverse Laplace transform:

• (s² + 3s + 8)Y(s) = (s + 5)X(s).

Taking the inverse Laplace transform, we get:

• y''(t) + 3y'(t) + 8y(t) = x'(t) + 5x(t)

This is the differential equation relating the output y(t) to the input x(t) for the specified system.