The following information applies to every part of this problem: A 1-kg mass hanging from a spring stretches it .3924 m in its equilibrium position. The spring is initially compressed to .5 m above the equilibrium point and released with an initial downwards velocity of 2 m/s. Use 9.81 m/s2 for the acceleration due to gravity. (a) Assuming that there is no damping and no additional force applied to the spring, write down and solve an initial value problem giving the position of the spring at time t. (b) Assuming that there is no damping but that an additional force of F(t) = 6 cos(4t) is applied to the spring, write down and solve an initial value problem giving the position of the spring at time t. (c) Compare the plots from parts (a) and (b). How did the plot change when an external force was applied? What stayed the same? (d) Assuming that there is a damping force proportional to the mass's velocity with pro- portionality constant 26 kg/s and no additional force applied to the spring, write down and solve an initial value problem giving the position of the spring at time t. Plot the result using and include a screenshot of your plot. (e) Assuming that there is a damping force proportional to the mass's velocity with pro- portionality constant 26 kg/s and that an additional force of F(t) = 6 cos(4t) is applied to the spring, write down and solve an initial value problem giving the position of the spring at time t. Plot the result using include a screenshot of your plot. (f) Compare the plots from parts (d) and (e). How did the plot change when an external force was applied? What stayed the same?