_ -wv----V- ~._ '_. .... . um. . um. U7 __..._._.-~ ... _ v-v-I-v-- 1. The differential equations 3:\" + 1033' +9$ = 0 and as\" + 61" + 9x = 0 both model springs. The difference between the two situations is the amount of friction; the coefcient of :5 (9 in this case) measures the \"springiness\" of the spring, and this is the same in both models. In this problem, you'll compare the two situations. (a) Find the solution of 1:" + 1037' + 99: = 0 satisfying the initial conditions 33(0) = 8 and $'(0) = 0. (b) Find the solution of 1;\" + 6m' + 93: = 0 satisfying the initial conditions m(0) = 8 and m'(0) = 0. (c) Hinged doors often have \"door closers\" which automatically swing the doors shut after they have been opened. Often, the door closer contains a spring that is at equilibrium when the door is shut. The door closer can be adjusted ( which essentially adjusts the amount of friction); in most cases, the goal is to adjust the door closer so that the door closes quickly but smoothly. Use a calculator or computer to graph the solutions you found in (a) and (b) together. Which one returns to 0 more quickly?(1) Is this the one with more or less friction? (Does this make sense to you intuitively?) (d) (Exploratory) What happens if you continue to adjust the friction in the \"better\" direction? (That is, if you thought more friction made the spring return to 0 more quickly, what happens if you increase the friction even more? If you thought less friction was better, what happens if you decrease the friction even more?) In particular, what can you say about the roots of the characteristic equation