Question

HW10-1. For a system with transfer function

H(s)= 2s + 3/s² + 2s + 5
a) For this system write the differential equation relating the output y(t) to the input x(t)

Answer 1

The differential equation relating the output y(t) to the input x(t) for the system with the transfer function H(s) is d²y/dt² + 2 · dy/dt + 5y = 2 · dx/dt + 3x.

Step-by-step explanation:

The student has asked to write the differential equation relating the output y(t) to the input x(t) for a system with the transfer function H(s) = (2s + 3)/(s² + 2s + 5). To do this, one needs to understand that the transfer function in the Laplace domain represents the ratio of the Laplace transform of the output to the Laplace transform of the input. Using the inverse Laplace transform, we can find the corresponding time-domain differential equation.

Assuming an arbitrary input x(t), its Laplace transform is X(s), and the transform of the output y(t) is Y(s). Given H(s) = Y(s) / X(s), we can multiply both sides by X(s) and the denominator of H(s) to obtain:

Y(s) · (s² + 2s + 5) = X(s) · (2s + 3)

Taking the inverse Laplace transform, the equation in the time domain becomes:

d²y/dt² + 2 · dy/dt + 5y = 2 · dx/dt + 3x

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