(Solved): (a) (5 marks) The input-output differential equation of a continuous-time system is given as follo ...

(a) (5 marks) The input-output differential equation of a continuous-time system is given as follows: \[ 2 \frac{d^{3} y(t)}{d t^{3}}+\frac{d^{2} y(t)}{d t^{2}}+\frac{1}{2} \frac{d y(t)}{d t}+y(t)=\frac{d x(t)}{d t}-x(t) \] Find the transfer function of the system. (b) (5 marks) For the system in (a), check the stability of the system using the Routh Hurwitz test. (c) (10 marks) Determine the Laplace Transform of the following signal: \[ x(t)=u(t)-2 \exp (3 t) u(t)+\cos (2 t) u(t) \] where \( u(t) \) is the unit-step function.