MATH 251 Examination I February 25, 2016 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination I February 25, 2016 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all 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. Do not write in this box. 1: (8) 2 through 8: (35) 9: (12) 10: (10) 11: (10) 12: (12) 13: (13) Total:

2 1. (8 points) Consider the following differential equation 4t 3 y t 2 e t + t 4 y = 0. (a) (2 points) What is the equation s order? Explain your answer. (b) (2 points) Is the equation linear? If yes, write it in the standard form. (c) (2 points) Is the equation separable? If yes, completely separate the variables. (d) (2 points) Is the equation exact? Why or why not? 2. (5 points) Consider the initial value problem (t 2 π 2 )y + (t 1)y = sin(2t), y( 2) = 5. According to the Existence and Uniqueness Theorem, what is the largest interval in which a unique solution is guaranteed to exist? (a) (π, ) (b) ( π, 1) (c) ( π, π) (d) (, ) Page 2 of 10

3 3. (5 points) What is a suitable integrating factor that can be used to solve the equation DO NOT solve this differential equation. 3(t 3 + 1)y + 3t 2 y = t 3? (a) µ(t) = 1 3 ln(t3 + 1) (b) µ(t) = 3 t (c) µ(t) = (t 3 + 1) 3 (d) µ(t) = e t3 4. (5 points) A 800 gallon tank initial contains 600 gallons of water and 60 pounds of dissolved salt. Brine solution enters the tank at the rate of 2 gal/min with a salt concentration of 4 lb/gal. The well mixed solution leaves the tank at the rate of 1 gal/min. Which initial value problem below models the amount of salt Q(t) inside the tank, as a function of time t, 0 t 200? (a) Q (t) = 8 (b) Q (t) = 8 (c) Q (t) = 8 (d) Q (t) = 8 Q(t), Q(0) = t Q(t), Q(0) = t Q(t), Q(0) = t Q(t), Q(0) = t Page 3 of 10

4 5. (5 points) Suppose y 1 (t) and y 2 (t) are two solutions of a certain second order linear differential equation sin(t)y cos(t)y + 2 ln(t)y = 0, 0 < t < π 2. Which function below represents the general form of their Wronskian, W (y 1, y 2 )(t)? (a) W (y 1, y 2 )(t) = Ce sin(t) (b) W (y 1, y 2 )(t) = C sin(t) (c) W (y 1, y 2 )(t) = C csc(t) (d) W (y 1, y 2 )(t) = Ce csc(t) 6. (5 points) Which equation below has the property that all of its nonzero solutions become unbounded as t? (a) 3y + 2y y = 0 (b) y y 4y = 0 (c) y + y + 4y = 0 (d) y 2y + y = 0 Page 4 of 10

5 7. (5 points) Suppose y 1 (t) = e t2 and y 2 (t) = e t+1 are two solutions of a certain second order differential equation y + p(t)y + q(t)y = 0. Which of the following statements is false? (a) y(t) = e t is also a solution. (b) y(t) = 1 2 et et2 is also a solution. (c) y(t) = e t+6 is also a solution. (d) y(t) = e t2 +t is also a solution. 8. (5 points) Find the general solution of the fourth order linear equation y (4) + 4y + 4y = 0. (a) y(t) = c 1 e 2t + c 2 e 2t + c 3 te 2t + c 4 te 2t (b) y(t) = c 1 cos( 2t) + c 2 sin( 2t) + c 3 t cos( 2t) + c 4 t sin( 2t) (c) y(t) = c 1 e 2t cos( 2t) + c 2 e 2t sin( 2t) + c 3 te 2t cos( 2t) + c 4 te 2t sin( 2t) (d) y(t) = c 1 e 2t cos( 2t) + c 2 e 2t sin( 2t) + c 3 te 2t cos( 2t) + c 4 te 2t sin( 2t) Page 5 of 10

6 9. (12 points) Consider the following list of differential equations: A. u + 8u + 16u = 0 B. u 7u + 6u = 0 C. u + u = 0 D. u + 4u + 5u = 2 E. u + 4u = 2 sin(2t) F. u + 8u + 7u = 0 G. u 9u = 4 sin(3t) Each of the equations above may or may not describe the displacement of a mass-spring system. Each question below has exactly one correct answer. The same equation may be reused to answer more than one question. (a) Which equation describes a mass-spring system that is critically damped? (b) Which equation describes a mass-spring system that is overdamped? (c) Which equation describes a mass-spring system that is undergoing resonance? (d) Which equation describes a mass-spring system that exhibits a simple harmonic motion? (e) Which equation describes a mass-spring system that has a quasi-frequency of 1 rad/sec? (f) Which equation describes a mass-spring system whose displacement always becomes unbounded regardless of initial conditions? Page 6 of 10

7 10. (10 points) Solve the following initial value problem. Give your answer in the explicit form. y = 2 et 2y 6, y(0) = 0. Page 7 of 10

8 11. (10 points) Consider the differential equation ( 4x 3 y 3 ye xy + x ) + ( 3x 4 y 2 xe xy y ) y = 0. (a) (3 points) Verify that it is an exact equation. (b) (7 points) Find the solution of this equation satisfying y(2) = 0. answer in an implicit form. You may leave your Page 8 of 10

9 12. (12 points) Consider the autonomous differential equation y = y 2 (y 2 6y + 8). (a) (2 points) Find all equilibrium solutions of this equation. (b) (6 points) Classify the stability of each equilibrium solution. Justify your answer. (c) (2 points) Suppose y(0) = α, and lim y(t) = 0. Find all possible value(s) of α. t (d) (2 points) Suppose y( 4) = β, and y(0) = 2. Find all possible value(s) of β. Page 9 of 10

10 13. (13 points) Consider the second order nonhomogeneous linear equation y 2y + 5y = 5t 2 + 6t 12. (a) (3 points) Find y c (t), the solution of its corresponding homogeneous equation. (b) (5 points) Find a particular function Y (t) that satisfies the equation. (c) (1 point) Write down the general solution of the equation. (d) (4 points) What is the form of particular solution Y that you would use to solve the following equation using the Method of Undetermined Coefficients? DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS. y 2y + 5y = e t (t 2 + t) cos(2t). Page 10 of 10