Math 250 Fall 2010 Final Exam NAME: ID No: SECTION: This exam contains 17 problems on 13 pages (including this title page) for a total of 150 points. There are 10 multiple-choice problems and 7 partial credit problems. In order to obtain full credit for the partial credit problems, all work must be shown. NO CALCULATORS, NOTES OR BOOKS ARE ALLOWED. All cellphones and music players, or any other electronic device must be put away. Problem Score Possible Points 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 10 8 11 7 12 9 13 13 14 14 15 18 16 20 17 7 Total 150
Multiple Choice Section 1. (6 points) Which of the following is the general solution of the equation: y +3y +2y = 0? (a) y = c 1 sin(2t)+c 2 cos(t); (b) y = c 1 e 2t +c 2 e t ; (c) y = c 1 e 2t +c 2 e t ; (d) None of the above. 2. (6 points) Which of the following second order differential equations is linear and homogeneous? (a) t 2 y +ty +(sint)y = 0; (b) (1+y 2 )y +ty +(sint)y = 0; (c) t 2 y +ty +(sint)y = lnt; (d) None of the above. Page 2 of 13
3. (6 points) Which of the following is the general solution of the equation: y y = e 3t? (a) y = t 3 +Ce t ; (b) y = te 3t +Ce t ; (c) y = 1 2 e3t +Ce t ; (d) y = 1 2 e3t +Ce t. 4. (6 points) Consider g(t) = t 2 3u 1 (t)+u 3 (t) tu 5 (t). What is g(3) g(0)? (a) 6 (b) 7 (c) 4+e 6 (d) 3 Page 3 of 13
5. (6 points) Find the solution of the initial value problem: (a) y = 2+ 3x e x +10; y = 3 ex 4+2y, y(0) = 1 MATH 250 Final Exam Fall 2010 (b) y = 2+ 3x 2 e x +10; (c) y = 4 3x e x +10; (d) y = 4 3x e x +10. 6. (6 points) Which of the following is the Laplace transform of f(t) = tu 3 (t)? (a) F(s) = e 3s ; (b) F(s) = e 3s s 2 ; (c) F(s) = e 3s ( 1 s 2 + 3 s ); (d) None of the above. Page 4 of 13
7. (6 points) Using the method of undetermined coefficients, a particular solution of y 4y +4y = te 2t +t 2 is of the form (a) (At+B)e 2t +Ct 2 +Dt+E; (b) (At 2 +Bt)e 2t +Ct 2 +Dt+E; (c) (At 3 +Bt 2 )e 2t +Ct 2 +Dt+E; (d) Ate 2t +Ct 2 +Dt+E. 8. (6 points) An interval, where the solution to the following initial value problem exists and is unique, is: (tsint)y +t 2 cost = y, y( 3) = 1. t 2 4 (a) (, 2); (b) ( 2, 2); (c) (2, ); (d) ( π, 2). Page 5 of 13
9. (6 points) Consider the system x = ( 0 1 1 0 ) x. Which of the following is the direction field of the above system? (A) (B) (C) (D) 10. (8 points) Consider the system x = ( 0 1 k 2 ) x. Which of the following statement is false: (a) The origin is a spiral point if k > 1 (b) The origine is a center when k = 1 (c) The origin is a saddle point when k = 3 (d) All saddle points of this system are unstable. Page 6 of 13
Partial Credit Section 11. (7 points) Find an integrating factor for the following differential equation: ty (t+2)y = 2tan(3t) 12 (9 points) Consider the autonomous differential equation y = y 2 (y 2 25). List the equilibrium points and determine their stability. No explanations required. equilibrium points stability Page 7 of 13
13. (13 points) Consider the differential equation t 2 y 3ty +4y = 0, t > 0. MATH 250 Final Exam Fall 2010 (a) (5 points) Use Abel s theorem to compute the Wronskian of a fundamental set of solutions of this differential equation without solving it. (b) (8 points) Use the Wronskian computed above to compute the general solution of the above differential equation, given that one solution is y 1 (t) = t 2. Page 8 of 13
14. (14 points) Consider the initial value problem: MATH 250 Final Exam Fall 2010 y +y = δ(t π), y(0) = 0, y (0) = 0. Use the Laplace transform to solve the given initial value problem by proceeding as follows. (a) (7 points) Find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem. (b)(7 points) Compute the inverse Laplace transform to solve the initial value problem. Page 9 of 13
15. (18 points) MATH 250 Final Exam Fall 2010 (a) (6 points) Transform the given system of differential equations below into a single equation of second order. (Do not solve it!) x 1 = 2x 1 3x 2 x 2 = x 1 x 2 (b) (6 points) Transform the given initial value problem for the single differential equation of second order into an initial value problem for two first order equations. (Do not solve it!) y +3y +y = e t, y(0) = 1, y (0) = 2. Page 10 of 13
(c) (6 points) (Problem 15, continued) ( ) 1 Consider the linear system x = Ax, where A is a 2 2 matrix with eigenvectors ( ) 1 1 and corresponding to the eigenvalues r 2 1 = 1 and r 2 = 2, respectively. Find a fundamental matrix Φ of the system such that Φ(0) = I. Page 11 of 13
16. (20 points) Consider the initial value problem ( 2 1 x = 1 4 MATH 250 Final Exam Fall 2010 ) x ; x(0) = (a) (4 points) Find the eigenvalue(s) and the eigenvector(s) of the above 2 2 matrix. ( 1 1 ). (b) (6 points) Find a fundamental set of two solutions for the system. (c) (5 points) Find the solution of the above initial value problem. (d) (5 points) Classify the origin and determine the stability of the system. Page 12 of 13
17. (7 points) (a) (3 points) Write down the equation that describes the Euler method for numerically approximating the solution to an initial value problem with step size h. dy dt = f(t,y), y(t 0) = y 0 Your answer: y n+1 = (b) (4 points) Consider the initial value problem dy dt = sint+y2 +1, y(0) = 1. Use Euler s method with step size h = 0.1 to find the approximate value of the solution of the initial value problem at t = 0.1. Page 13 of 13