MATH 251 Examination I October 10, 2013 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination I October 10, 2013 FORM A Name: Student Number: Section: This exam has 13 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. Do not write in this box. 1 through 9: (45) 10: (14) 11: (13) 12: (13) 13: (15) Total:

2 1. (5 points) Which of the equations below is a second order nonlinear differential equation? (a) t 2 s (1 s ) 2 = e 2t (b) e t y sin(t)y = tan(t) (c) ty 5 sin(2t)y = e y (d) t 3 r e 3 cos(t) r = t 4 2. (5 points) What is a suitable integrating factor for solving the follwing linear equation ty + 3 y = cos t, t > 0? 2 (a) µ(t) = t 3 2 (b) µ(t) = 3t 2 (c) µ(t) = 3 2 ln(t) (d) µ(t) = e 3t 2 Page 2 of 9

3 3. (5 points) Solve the initial value problem y = xe 2x, y(0) = 1. 2y 4 (a) y = xe 2x e 2x + 5 (b) y = xe 2x e 2x + 19 (c) y = 2 + (d) y = xe 2x 1 4 e 2x xe 2x e 2x (5 points) Consider all nonzero solutions of the linear equation y + 4y 5y = 0. As t, they will (a) all approach 0. (b) some approach, all the rest approach. (c) some approach 0, some approach, some approach. (d) neither approach any limit, nor approach ±. Page 3 of 9

4 5. (5 points) A tank with capacity 500 gal initially contains 400 gal of water with 40 lb of salt in solution. Water containing sin(30t) lb of salt per gallon is entering at a rate of 3 gal/min, and the mixture is allowed to flow out of the tank at a rate of 1 gal/min. Let Q(t) denote the amount of salt in the tank in time t. Which of the following initial value problems accurately describes Q(t) for 0 < t < 50? (a) dq dt (b) dq dt (c) dq dt (d) dq dt = 3( sin(30t)) + Q(t), Q(0) = t Q(t) = 3( sin(30t)), Q(0) = = 3( sin(30t)) Q(t), Q(0) = t = ( sin(30t)) 3 Q(t), Q(0) = t 6. (5 points) What is the general solution of the following equation cos(x + y) + 2x + (cos(x + y) + 4y)y = 0? (a) cos(x + y) + x 2 = C (b) sin(x + y) + y 2 = C (c) sin(x + y) + 2x + 4y = C (d) sin(x + y) + x 2 + 2y 2 = C Page 4 of 9

5 7. (5 points) Consider the differential equation (2x + αy + 2xe x2 ) + (2y + x) dy dx = 0. For what value of α will the equation be an exact equation? (a) α = 3 (b) α = 1 (c) α = 0 (d) The equation is exact for any value of α. 8. (5 points) The velocity, given in meters per second, of a certain particle is given by the initial value problem dv dt = v2, v 0, v(0) = 5. Approximately how fast will the particle be moving after a very long time? Hint: the answer can be deduced without having to solve the initial value problem. (a) 25 m/s (b) 200 m/s (c) m/s (d) Page 5 of 9

6 9. (5 points) Consider the initial value problem t(t 2)(t 4)y + y = e t2, y(3) = 6. What is the largest interval in which a unique solution is guaranteed to exist? (a) (4, ) (b) (, 0) (c) (2, 4) (d) (, ) 10. (14 points) A mass-spring system is described by the equation 5u + γu + 80u = 0. (a) (3 points) When γ = 0, what is the system s natural period? (b) (5 points) When γ = 40, find the displacement u(t) that satisfies u(0) = 6 and u (0) = 4. (c) (3 points) When γ = 40, is the system underdamped, overdamped, or critically damped? (d) (3 points) True or False: Regardless of the initial conditions, no solution of the system described in (b) can cross the equilibrium position more than once. Page 6 of 9

7 11. (13 points) Consider the autonomous differential equation y = (y 1) 2 (y 2 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(1325) = 1.5, then what is lim t y(t)? (d) (2 points) If y( 1325) = 2, what is lim t y(t)? Page 7 of 9

8 12. (13 points) Consider the second order nonhomogeneous linear equation y + 3y + 2y = 20 sin(2t) + 4t. (a) (3 points) Find y c (t), the solution of its corresponding homogeneous equation. (b) (8 points) Find its general solution. (c) (2 points) Find the solution satisfying the conditions y(0) = 0 and y (0) = 2. Page 8 of 9

9 13. (15 points) Given that y 1 (t) = 1 and y 2 (t) = arctan(t) are both solutions of the second order homogeneous linear equation (t 2 + 1)y + 2t y = 0. Determine whether each of the following statements is true or false. State a brief reason that justifies each answer. (a) (3 points) Wronskian W (y 1, y 2 )(t) = 0. (b) (3 points) y 1 and y 2 form a set of fundamental solutions of this equation. (c) (3 points) y 3 (t) = arctan(t) is also a solution of the equation. (d) (3 points) There are additional solutions that cannot be expressed as a linear combination of y 1 and y 2. (e) (3 points) Each solution is unique to its corresponding initial conditions only on the interval 1 < t < 1. Page 9 of 9