First put the following second$-$order differential equations in the standard canonical transfer func$-$

tion form K$2$

n

s$2+2$ns$+$$2$

n

$.$ I$)$ Determine the damping ratio, natural frequency, DC gain, time constant

and the settling time of each system. II$)$ Determine if the system is overdamped, underdamped

or critically damped. If it is underdamped calculate the frequency of oscillation $($d$)$ and the

maximum overshoot of the system.

$($a$)4$x $+12$x $+36$x $=40$u$($t$)$

$($b$)$x $+9$x $+9$x $=9$u$($t$)$

$($c$)2$x $+8$x $+8$x $=80$u$($t$)$

$($d$)2$x $+2$x $+200$x $=10$u$($t$)$