a$)$ Write the differential equation for this system, relating the input with the output and showing every step of your work. Then write the equation with the substitution of the element values $R=2,$ and $C=0\mathrm{.}1F.$

b$)$ Using the Laplace Transform, find the solution for the differential equation found in a$),$ given that the input is $x(t)=e^{-4t}u(t)$ and that at time $t=0,$ the initial capacitor voltage is $y(0)=1V.$ Show your step$-$by$-$step solution. Comment briefly on why it is appropriate to use the Laplace Transform to solve this LTI system $($advantages$,$ disadvantages, use of unilateral LT$,$ etc$)$

c$)$ Plot the solution using the code 'Plotting the system response code' included on Week $11.$ Comment on the changes made to the original code $($e$.$g$.$ the axis range$)$ and on the characteristics of the response.