Problem Score Possible Points Total 150

Transcription

1 Math 250 Spring 2010 Final Exam NAME: ID No: SECTION: This exam contains 17 problems on 14 pages (including this title page) for a total of 150 points. The exam has a multiple choice part, and 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 cell phones and music players must be put away. Problem Score Possible Points Total 150

2 Multiple Choice Section MATH 250 Spring 2010 Final Exam 1. (5 points) Suppose y 1 (t) and y 2 (t) are solutions of y +p(t)y +q(t)y = g(t). Which of the following functions is also a solution of y +p(t)y +q(t)y = g(t)? (a) y 3 = y 1 +y 2 ; (b) y 3 = y 1 2y 2 ; (c) y 3 = y 2 y 1 ; (d) y 3 = 3y 2 2y (4 points) The critical point of the system is x 1 = x 1 +2x 2 x 2 = 5x 1 x 2 (a) a node ; (b) a spiral point; (c) a center; (d) a saddle point. Page 2 of 14

3 3. (5 points) Consider a spring mass system described by 4u +3u +u = 0. Which of the following is NOT true? (a) The equation describes a damped system with mass 4 and spring constant 1. (b) The system has quasi period 16π/3. (c) Irrespective of the initial conditions, the system will eventually come to rest. (d) The quasi frequency is 7/8. 4. (5 points) Find the inverse Laplace transform of the function F(s) = 3s 2 s 2 2s (a) 2e 2t +1. (b) 2e 2t 1; (c) e 2t +1; (d) e 2t 1; Page 3 of 14

4 5. (5 points) Find the Laplace transform of the function f(t) = u 2 (t)e 5t MATH 250 Spring 2010 Final Exam (a) e 2s+10 1 s 5 ; (b) e 2s 1 s+5 ; (c) e 2s 1 s 5 ; (d) e 2s 10 1 s (5 points) Consider f(t) = t (tu 2 (t) u π (t))(u 5 (t) tu 7 (t)). What is f(6)? (a) 1 ; (b) 29; (c) 54 5π; (d) 30+5π. Page 4 of 14

5 7. (6 points) Consider the autonomous differential equation y = (y+1) 2 (y 5). List the critical points and determine their stability. No explanations required. critical points stability 8. (5 points)considerthedifferentialequationy 2y = e 2t withinitialconditiony(0) = 2. Which of the following is the solution to this initial value problem: (a) y(t) = e 2t +e 2t ; (b) y(t) = (t+2)e 2t ; (c) y(t) = (t+2)e 2t ; (d) y(t) = 2e 2t e 2t. Page 5 of 14

6 9. (5 points) Using the method of undetermined coefficients, a particular solution of y 6y +9y = te 3t +t 2 3 is of the form (a) (At+B)e 3t +Ct 2 +Dt+E; (b) (At 2 +Bt)e 3t +Ct 2 +Dt+E; (c) Ate 3t +Ct 2 +Dt+E; (d) (At 3 +Bt 2 )e 3t +Ct 2 +Dt+E. 10. (5 points) Find the general solution of the following differential equation: x 2 y = y 2. (a) y(x) = Cx; (b) y(x) = 1 1 x +C; (c) y(x) = Ce 1/x2 ; (d) y(x) = C 1 sin(x)+c 2 cos(x). Page 6 of 14

7 11. (5 points) Consider the system x = Φ(0) = I is given by (a) 1 2 e3t 1 2 e t 1 4 e3t 1 4 e t e 3t e t 1 2 e3t 1 2 e t ( MATH 250 Spring 2010 Final Exam ) x. A fundamental matrix Φ such that (b) 1 2 e3t e t 1 4 e3t 1 4 e t e 3t e t 1 2 e3t e t (c) 1 2 e 3t 1 2 et 1 4 e 3t 1 4 et e 3t e t 1 2 e 3t 1 2 et (d) 1 2 e 3t et 1 4 e 3t 1 4 et e 3t e t 1 2 e 3t et Page 7 of 14

8 Partial Credit Section MATH 250 Spring 2010 Final Exam 12. (14 points) (a) (7 points) Transform the given system of differential equations below into a single equation of second order with proper initial conditions. (Do not solve it.) x 1 = x 1 x 2 x 1 (0) = 1 x 2 = 3x 1 4x 2 x 2 (0) = 2. (b) (7 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.) u +1/4u +4u = 2cos(3t) u(0) = 1,u (0) = 2. Page 8 of 14

9 13. (20 points) Consider the initial value problem ( 1 4 x = 4 7 ) x ; x(0) = MATH 250 Spring 2010 Final Exam ( 2 1 Find the general solution of the system, and find the solution of the initial value problem. Also classify the origin and determine the stability of the system. ). Page 9 of 14

10 14. (20 points) ( ) 1 5 (a) (5 points) Consider the system of equations x = x. 1 3 Find the eigenvalues for the given system of equations. Classify the origin and determine the stability of the system. (You don t have to find the general solution.) (b) (5 points) Draw the phase portrait for the above system. Page 10 of 14

11 (c) (10 points) (Problem 14, continued) ( ) 3 0 Consider the system of equations x = x. 0 5 Find the eigenvalues for the given system of equations. Classify the origin and determine the stability of the system. Draw the phase phase portrait for this system. Page 11 of 14

12 15. (18 points) Consider the differential equation t 2 y ty +y = 0, t > 0. (a) (6 points) Use Abel s theorem to compute the Wronskian of a fundamental set of solutions of this differential equation without solving it. (b) (12 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. Page 12 of 14

13 16. (15 points) Consider the initial value problem: y +4y = δ(t 4π), y(0) = 1/2, y (0) = 0. Use the Laplace transform to solve the given initial value problem by proceeding as follows. (Throughout, label the identities you are using by referring to the appropriate line from the table of Laplace transforms.) (a) (6 points) Find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem. (b)(9 points) Compute the inverse Laplace transform to solve the initial value problem. Page 13 of 14

14 17. (8 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) (5 points) Consider the initial value problem dy dt = 3 2t 1 y, y(0) = 1. 2 Use Euler s method with step size h = 0.2 to find the approximate value of the solution of the initial value problem at t = 0.2. Page 14 of 14