Use Laplace Transform to find the response x(t) of the system governed by the following equation, where y(t) is the input: \frac{d^{2} x(t)}{d t^{2}}+5 \frac{d x(t)}{d t}+4 x(t)=6.5 \frac{d y(t)}{d t}+4 y(t) \text { Assume zero output initial conditions i.e. } x(0)=0, \dot{x}(0)=0 \text { and given that: } y(t)=2 \text { for } t \geq 0 The following equations represent a system, having an input r(t) and an outputv(t). Put these equations in transfer function form. \frac{d^{2} v(t)}{d t^{2}}+6 \frac{d v(t)}{d t}+5 v(t)=7 \frac{d r(t)}{d t}+3 r(t) c) A feedback system is shown in Figure Q1(c) below.

(i)Find the closed-loop transfer function: T(s)=\frac{y(s)}{u(s)} \text { when } G(s)=\frac{8}{s(10+0.5 s)} Calculate the values of the system poles. i) Using the final value theorem with unit step input, determine the steady-state value of y(t). Determine the overshoot, the rise time and the settling time of theresponse for unit step input.