Short answer

a$.$ Give a differential equation whose transfer function is given as follows $($where the

output is $x(t),$ the input is $f(t)$ :

$T(s)-=\frac{x(s)}{F(s)}=\frac{3s+1}{2s^2+3}$

b$.$ For the following dynamic system, what is the approximate time for the transient

response to decay to within $2\%$ of its initial value?

$2x^{}+18x^{}+50x=f(t)$

c$.$ What is the damping ratio $$ for the dynamic system in part $($b$)?$

d$.$ For each of the following systems, determine the roots, "sketch" the free response and

state the stability of the system $($stable or unstable$).$

i$.x^{}+6x^{}+90x=f(t)$

ii$.x^{}+6x^{}+25x=f(t)$