The following equation describes a second-order system: 

$$(d^2y)/dt^2+3dy/dt+36y=x(t)$$

The system may be described as:

A. nonlinear 

B. overdamped 

C. critically damped

D. underdamped

The characteristic equation of second order systems is as follows:

$m\left(\frac{d^{2}y}{d{t}^2}\right)+b\left(\frac{dy}{dt}\right)+k\left(y\right) = 0$

For the second order system $\frac{d^{2}y}{d{t}^2}+3\left(\frac{dy}{dt}\right)+36y = x\left(t\right)$, we write the characteristic equation as:

$\frac{d^{2}y}{d{t}^2}+3\left(\frac{dy}{dt}\right)+36y = 0$

From the characteristic equation, we can calculate the damping ratio which describes how oscillations in the system decay following a disturbance. The damping ratio is represented by $\zeta$ and is calculated based on the equation:

$\zeta = \frac{b}{2\sqrt{km}}$

By plugging in the coefficients to the equation for the damping ratio, we find that our damping ratio is equal to ¼:

$\zeta = \frac{3}{2\sqrt{\left(36\right)\left(1\right)}}=\frac{3}{2\sqrt{36}}=\frac{3}{12}=\frac{1}{4}$

Based on the value of the damping ratio $\zeta$:

1. A system is described as undamped if $\zeta = 0$
2. A system is described as underdamped if $0<\zeta <1$
3. A system is described as critically damped if $\zeta = 1$
4. A system is described as overdamped if $\zeta > 1$

Because $0 < \zeta =\frac{1}{4}<1$, the system may be described as underdamped (D).