MATH 251 Examination I July 1, 2013 FORM A Name: Student Number: Section: This exam has 12 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: 2 thru 6: 7: 8: 9: 10: 11: 12: Total:
1. (5 points) Give an example of the following: a) A first order, linear, autonomous, ordinary differential equation. b) A second order, non-linear, ordinary differential equation. 2. (5 points) Which of the following is an integrating factor for the differential equation (x 2 1)y = 2xy + x, x > 1? (a) µ(t) = e x2 (b) µ(t) = x 2 1 (c) µ(t) = e x2 (d) µ(t) = ln (x 2 1) Page 2 of 10
3. (5 points) Consider the initial value problem (t 2 9)y (2t sin t)y = ln(1 t), y(0) = 4. According to the Existence and Uniqueness Theorem, what is the largest interval in which a unique solution is guaranteed to exist? (a) ( 3, 3) (b) (3, ) (c) (1, 3) (d) ( 3, 1) 4. (5 points) Consider the equation: (x 3 y 2 + βxe y + ln x) + ( 1 2 x4 y 6y sin y 2 + x 2 e y )y = 0. Find the value of β such that the above equation is exact. (a) β = 1 (b) β = 1 (c) β = 2 (d) β = 2 Page 3 of 10
5. (5 points) Suppose that y 1 (t), y 2 (t) are two fundamental solutions of the equation ty 6y + 5e t y = 0 such that y 1 (2) = 8, y 1 (2) = 0, y 2(2) = 4, y 2 (2) = 8. Compute the Wronskian W (y 1, y 2 )(t) as a function of t. (a) t 6 (b) 32t 6 (c) 6 ln t (d) 2t 6 6. (5 points) A tank is filled with 100 liters of a solution containing 10 grams of salt. A solution containing a concentration of 3 g/liter of salt enters the tank at the rate 4 liters/minute and the well-stirred mixture leaves the tank at a rate of 5 liters/minute. Which of the initial value problems below describes, Q(t), the amount of salt in the tank at time t > 0? (a) Q = 12 1 20Q, Q(0) = 100 (b) Q = 12 5 100+tQ, Q(0) = 10 (c) Q = 12 5 100 tq, Q(0) = 100 (d) Q = 12 5 100 tq, Q(0) = 10 Page 4 of 10
7. (10 points) y 1 (t) = te t and y 2 (t) = t 2 are both solutions of a second order linear differential equation y + p(t) y + q(t) y = 0 (a) (4 points) Compute W (y 1, y 2 )(t). (b) (2 points) (TRUE or FALSE) y 1 and y 2 is not a fundamental pair of solutions. (c) (2 points) (TRUE or FALSE) y = 0 is a solution of the equation. (d) (2 points) (TRUE or FALSE) y = (10t 4e t )t is not a solution of the equation. Page 5 of 10
8. (12 points) Given that y 1 (t) = t 1 is a solution to the equation, t 2 y + 3ty + y = 0, t > 0, (a) (7 points) Use the method of reduction of order to find another solution y 2 which is not a scalar multiple of y 1. (b) (2 points) Find the general solution of the equation. (c) (3 points) Find a solution satisfying the following initial conditions: y(e) = 1 e, y (e) = 1 e 2. Page 6 of 10
9. (12 points) Consider a second order linear differential equation 3y 15y + 18y = 0 (a) (4 points) Find the general solution of the equation. (b) (5 points) Find the solution satisfying y(0) = 0, y (0) = α. (c) (3 points) For what value(s) of α is the lim t y(t) = 0? Page 7 of 10
10. (13 points) Consider the autonomous differential equation y = 36y 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(1984) = 0, what is y(2013)? Without solving the equation, briefly explain your conclusion. (d) (2 points) If y(t) is a solution that satisfies y(0) = 0.00001, then what is lim t y(t)? Page 8 of 10
11. (13 points) Consider the second order non-homogeneous linear equation y + 4y + 4y = 3e t t. (a) (3 points) Find y c (t), the solution of its corresponding homogeneous equation. (b) (7 points) Find the general solution of the equation. (c) (3 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 + 4y + 4y = t 2 cos t e 2t + 5. Page 9 of 10
12. (10 points) A mass-spring system is described by the equation 2u + γu + ku = F (t). (a) (2 points) Suppose the mass originally stretched the spring 4 meters to reach its equilibrium position. What is the spring constant k? (Assume g = 10 m/s 2 to be the gravitational constant.) (b) (2 points) Suppose k = 0.5, and F (t) = 0. For what value(s) of γ would this system be critically damped? (c) (2 points) Suppose γ = 0, k = 10, and F (t) = 0. What is the natural frequency of this system? (d) (2 points) True or false: Suppose γ = 0, k = 2, and F (t) = 5 cos t, then the mass-spring system is undergoing resonance? (e) (2 points) Suppose γ = 2, k = 1, F (t) = 0. What is the quasi-frequency of the system? Page 10 of 10