A spring would be stretched to a maximum of $6cm$ when a $4-$N weight is attached to its free end. Let the $4-$N weight be attached to the spring and pull down $4cm$ below the equilibrium position. If the weight is started with an upward velocity of $0\mathrm{.}02\frac{m}{s},$ determine the following:

a$.$ The governing differential equation of the system described above. Note that the coefficient of the $y^{\text{'}\text{'}}$ is mass, and the coefficient of $y$ is the value of weight over its maximum stretched value. Use SI units.

b$.$ Find the equation of the displacement $(y(t)),$ noting that the downward motion direction is set to be positive.

c$.$ Find the displacement equation $(y(t))$ if the system described above has an additional damping element with a damping coefficient of $0\mathrm{.}6N-\frac{s}{m}.$

d$.$ Compare the displacement and acceleration value obtained from item $b$ and item c when the time is $2$ seconds.