For the differential equation given below:

$\frac{d^{2}y(t)}{d{t}^2}+7\frac{dy(t)}{dt}+10\mathrm{.}81y(t)=12x(t)-7$

a$)$ Obtain $Y(t)$ as a deviation from its initial steady$-$state. Use the method of Laplace transforms and partial fractions expansion. The forcing function is $x(t)=u(t).$

b$)$ According to your solution predict if the response is:

monotonic or oscillatory

stable or instable

c$)$ Find the final steady$-$state value of the output.

d$)$ If the response is monotonic find the dominant root and its corresponding $t_k.$ If the response is oscillatory find the period, the decay ratio, and the settling time.