(Solved): (a) Solve the initial value problem using Laplace Transform and Inverse Laplace Transform \[ \frac{ ...

(a) Solve the initial value problem using Laplace Transform and Inverse Laplace Transform \[ \frac{d^{2} y(t)}{d t^{2}}-6 \frac{d y(t)}{d t}+15 y(t)=\frac{d x(t)}{d t}, \quad y(0)=-1, \frac{d y(0)}{d t}=-4 \] where \( x(t)=\int_{0}^{t} 2 \sin (3 t) d t \). (10 points) (b)Write the poles of the system given by the equation above. Draw the Region of Convergence (ROC). Is the system stable? Why?(3 points) (c)Let \( s=\sigma+j \omega \), can you find the Fourier Transform of the system above for \( \omega=0 \) ? Why? If it can be evaluated, then derive Fourier Transform of the response of the system above. (2 points) Solution: