The differential equation relating the output y(t) to the input x(t) is (dy/dt) - 8y = 0. This can be obtained by converting the transfer function to a polynomial equation, setting the denominator to zero, and writing the corresponding differential equations.
Convert the transfer function to a polynomial equation:
Multiply both sides of the equation by the denominator, D(s) = s² + 8s² + 5s + 7:
s² + 3s + 5 = (s² + 8s² + 5s + 7)y(t)
Set the denominator to zero and solve for s:
s² + 8s² + 5s + 7 = 0
Factor the polynomial:
(s + 7)(s + 1) = 0
Solve for the roots:
s = -7, -1
Write the characteristic equation:
(s + 7)(s + 1) = 0
Write the corresponding differential equations:
For each root, write a differential equation of the form:
(dy/dt) + p_i y = 0
For s = -7:
(dy/dt) - 7y = 0
For s = -1:
(dy/dt) - y = 0
Combine the differential equations:
Since the system is assumed to be controllable and observable, the complete differential equation can be obtained by combining the two differential equations:
(dy/dt) - 7y - y = 0
(dy/dt) - 8y = 0
Therefore, the differential equation relating the output y(t) to the input x(t) is:
(dy/dt) - 8y = 0