MATH 251 Examination I July 5, 2011 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination I July 5, 2011 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit problems, all work must be shown. Credit will not be given for an answer not supported by work. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of work. The point value for each question is in parentheses to the right of the question number. You may not use a calculator on this exam. Please turn off and put away your cell phone and all other mobile devices. Do not write in this box. 1 : 2 thru 6: 7: 8: 9: 10: 11: 12: Total:

2 1. (8 points) Consider the differential equation yy = 1 x 2. Answer the following questions. You must give a reason to justify each of your answers. (a) (2 points) Is the equation first order linear? (b) (2 points) Is the equation separable? (c) (2 points) Is the equation exact? (d) (2 points) Is the equation autonomous? Page 2 of 10

3 2. (5 points) Solve the initial value problem y = 4x sin x, y(0) = 1. y 2 (a) y = 2 + 4x cos x 1 (b) y = 2 4x cos x 1 (c) y = 2 + 2x 2 cos x + 1 (d) y = 2 2x 2 cos x (5 points) Consider the initial value problem (t 2 + 4t)y + t + 3 2t 4 y = t3 e t sin(4t), y( 2) = π. What is the largest interval in which a unique solution is guaranteed to exist? (a) (, 2) (b) ( 4, 0) (c) (0, 2) (d) (2, ) Page 3 of 10

4 4. (5 points) Let y(t) be the solution of the initial value problem y + 3y 10y = 0, y(0) = α, y (0) = 2. Suppose lim t y(t) = 0, find the value of α. (a) 1 (b) 10 (c) 2 5 (d) (5 points) Let y 1 (t) and y 2 (t) be any two solutions of the second order linear equation (t + 2)y + 4y + (t 2 + 9)e 5t y = 0. What is the general form of their Wronskian, W (y 1, y 2 )(t)? (a) Ce 4t (b) Ce 4t (c) C(t + 2) 4 (d) C (t + 2) 4 Page 4 of 10

5 6. (5 points) Which of the following second order linear equations below has y = 6e 3t 3e t as a solution? (a) y + 3y = 0 (b) y + 2y 3y = e 3t sin t (c) 2y 4y 6y = 0 (d) y 2y + 3y = 0 7. (9 points) A mass weighing 0.25kg stretches a spring 1.25m The mass-spring system has a damping constant of 1 2kg/s. At t = 0 the mass is displaced an additional 50cm downward from its equilibrium position and set in motion with an upward velocity of 2m/s. (You may use g = 10m/s 2 as the gravitational constant.) (a) (5 points) Write an initial value problem that describes the motion of the mass. (b) (4 points) Determine the system s quasi frequency and quasi period. Page 5 of 10

6 8. (11 points) Consider the second order linear equation t 2 y 3ty + 4y = 0, t > 0. (a) (2 points) Verify that y 1 (t) = t 2 is a solution of this equation. (b) (9 points) Find the equation s general solution using the method of reduction of order. Page 6 of 10

7 9. (13 points) Consider the autonomous differential equation y = y(1 y) 2 (y + 4). (a) (3 points) Find all of its equilibrium solutions. (b) (6 points) Classify the stability of each equilibrium solution. Justify your answer. (c) (2 points) If y(9π) = 2, then what is lim t y(t)? (d) (2 points) If y( 50) = 1, then what is y(2)? Page 7 of 10

8 10. (12 points) Consider the equation 3x 5 y 4 2x + (2x 6 y 3 + 6e 3y ) y = 0. (a) (3 points) Verify that it is an exact equation. (b) (7 points) Find its general solution. You may leave your answer in implict form. (c) (2 points) Find the particular solution satisfying the initial condition y(3) = 0. Page 8 of 10

9 11. (11 points) A fish tank initially contains 800 grams of oxygen dissolved in 200 liters of water. Water containing 8 grams/liter of oxygen flows into the tank at a rate of 2 liters/min. The well-stirred mixture flows out of the tank at the same rate. (a) (4 points) Set up an initial value problem (that is, give both a differential equation and an initial condition) modeling this process that describes the amount of oxygen present in the tank at any time t > 0. (b) (5 points) Solve the initial value problem thus obtained. (c) (2 points) What will the concentration of oxygen in the fish tank be as t? Page 9 of 10

10 12. (11 points) Consider the nonhomogeneous second order linear equation of the form y 8y + 16y = g(t). (a) (2 points) Find its complementary solution, y c (t). (b) (6 points) Find the general solution of y 8y + 16y = 10e 2t (c) (3 points) What is the correct form of particular solution that you would use to solve the equation below using the Method of Undetermined Coefficients? DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS. y 8y + 16y = 9 cos 4t 5t 2 e 4t Page 10 of 10