NAME NetID MATH 285 E1 F1 Exam 2 A October 17 2014 Instructor Pascaleff Problem Possible Actual 1 20 Do all work on these sheets 2 20 Show all work 3 20 The exam is 50 minutes 4 20 Do not discuss this exam with anyone until after 3 00 pm on Oct 17 2014 5 20 Total 100 INSTRUCTIONS 1 1 20 points Let P t denote the population of a penguin colony in Antarctica We assume that each female lays one egg each year so the birth rate is 0 5 births per penguin per year Due to scare resources the death rate depends on the population as 05 001P deaths per penguin per year Suppose that the penguin population is in equilibrium meaning that it is constant in time P t P0 What are the possible values of the equilibrium constant population P0 2 2 2 2 20 points Show that the functions f x ex 5 and g x e3x and h x 4ex are not linearly independent That is find constants A B C such that Af x Bg x Ch x 0 for all x 3 3 20 points For each polynomial differential operator p D find the solutions to the homod geneous differential equation p D y 0 where D dx It is not necessary to rederive the solution completely but keep in mind that partial credit can be given if substantial work is shown In the first three parts you are asked to find the general real not complex solution In the last part you are asked to find one complex solution a p D D2 3D 2 Find the general real solution of p D y 0 b p D D 4 D 3 3 Find the general real solution of p D y 0 4 c p D D2 D 4 Find the general real solution of p D y 0 d p D D 3i where i 1 In this case there are no real solutions Find one complex solution of p D y 0 5 4 20 points A mass is attached to a spring and a dashpot so that its position x t obeys the differential equation d2 x dx m 2 c kx 0 dt dt The mass m damping coefficient c and spring constant k are given by m 3 c 6 k 3 Suppose we do an experiment where the initial position and the initial velocity are x 0 1 and v 0 10 The function x t is determined by these initial conditions plus the differential equation How many times does the mass pass through the equilibrium position x 0 That is how many positive numbers t are there such that x t 0 Note that if the solutions oscillate the answer could be infinitely many 6 5 20 points Find the general solution of the differential equation y 00 3y 0 2y 2x 7
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