Name: MA 226 FINAL EXAM Show Your Work and JUSTIFY Your Responses. Clearly label things that you want the grader to see. You are responsible for conveying your knowledge of the material in an understandable fashion. Problem Possible Actual Score 1 36 2 24 3 24 4 24 5 24 6 24 7 24 TOTAL 180
1.) 20 points - Short Answer (4 each) A) A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing 5 parts per billion of dioxin flows into the top of the tank at a rate of 4 gallons per minute. The water in the tank is kept well mixed, and 2 gallons per minute are removed from the bottom of the tank. To find the general solution to the differential equation modeling the amount of dioxin in the tank, what method from Chapter 1 involving the product rule would you need to use? How do you know you ll need it? B) Locate the bifurcation value(s) of α for dy dt = y6 2y 3 + α C) Find the equilibrium solutions for the differential equation dy dt = (y2 2)(y + 1)(t 3) (5 y 2 )(7 + t)
D) Below is the result of Euler s Method, from 0 t 5, where t = 0.5, for the IVP: dy dt = (3 y)(y + 1), y(0) = 0. What is wrong with this estimate given by Euler s Method? What is a potential fix for this? E) Say your friend was using Euler s Method for systems on the system ( ) dy 0 1 dt = Y. 1 0 You know the solution to the system is < sin(t), cos(t) >. You also know that Euler s Method won t work out well for your friend. Explain why to your friend. What is a potential fix for this? F) Your friend wants to plug a system into HPGSystemSolver or another graphing program so that a particular solution curve looks like the symbol centered at the origin. What must be true about this system for that to be possible?
G) If the nonhomogenous portion of a second-order linear DE is: e 3t cos 4t with corresponding complexified e ( 3+4i)t and the required guess for the particular solution is Re[αte ( 3+4i)t ] or te 3t (α cos(4t) + β sin(4t)), what must the left hand side of the DE be? H) Write down a function using the Heaviside function u a (t) such that 2, if 0 t < 5 g(t) = 7, if 5 t < 11 e 3(t 11), if 11 t I) You are an engineer working for a company that designs the spring thing that is connected to the tops of office doors so they close automatically. Your boss (not an engineer) tells you that a market research survey indicates that most people want doors that swing twice. I.e., after a door is opened, the door would pass through the closed position three times and eventually close from the side the person was not on. What do you tell your boss?
2.) Find the solution for y + 2y + y = 2 cos(2t) with y(0) = 1, y (0) = 0
3.) For each second-order equation, classify each harmonic oscillator as critically damped, overdamped, underdamped, undamped + resonant, or undamped + beating. Then sketch a typical solution in the yt-plane. The y-axis does not need a scale, but the t-axis does. a) y + 100y + 9y = cos(3t) (don t use y(0) = y (0) = 0 for this one) b) y + 9y = cos(3t) c) y + 4y = cos(1.75t) d) y + 25y = cos(5.25t)
4.) Consider dx dy = x(10 x y) and dt dt a) Find all equilibrium points. b) Classify each equilibrium point. = y(30 2x y)
5.) Consider dx dy = x(10 x y) and = y(30 2x y) dt dt We restrict attention to the first quadrant. a) Find and sketch the nullclines. b) Indicate any equilibrium points on your sketch. c) Find the directions on the nullclines and indicate those directions on your sketch with arrows. d) Sketch solution curves in the xy-plane for the initial values (3,0), (0, 3), (5,5), and (30, 30). e) Briefly describe the possible behaviors of solutions.
6.) Eric s soul is a collection of experiences. Every time a student asks him about his office hours instead of using the syllabus, he experiences rage. Every time a student asks him in lecture a question that he literally exactly answered 10 seconds ago, this is actually totally fine. On the other hand, every time a student asks him if something will be on the exam, he experiences RAGE. Eric s soul is probably infinite, but for the purposes of this problem it can contain 1000 experiences. A typical day averages out to the soul taking in 40 experiences, with 11 of those experiences being rage-inducing questions. Eric s soul is well mixed. Eric s soul sheds experiences so that it s constantly full at 1000 experiences. Eric s soul was pristine with only peaceful experiences before the semester began. a) What is the salt here? What is the salt water? What is the x-gallon vat? b) Identify variables and units. c) Write an initial value problem modeling the rate of change of rage in Eric s soul per day. d) Find the particular solution for the amount of rage in Eric s soul at a given time t. e) Eric writes ridiculously angry and difficult exams if his soul is over 25% rage. Given that the semester is easily long enough for his soul to reach equilibrium rage levels, what does this tell you about the final exam? f) Is there anything you d like to say about this problem?
7.) a) Solve for L[y] given y + 4y + 7y = 13t 2 δ 5 (t) + 8u 3 (t)e 4(t 3), y(0) = 2, y (0) = 1 b) Solve for y(t) given ( 2 L[y] = e 5s s + 3 s 7 + 2s + 4 s 2 + 6 + 2s + 4 ) s 2 + 4s + 6 + 2 s + 2s + 4 s 2 + 4s + 6