midterm2_2007_spring

midterm2_2007_spring - Name: MATH 250 Student Number:...

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Unformatted text preview: Name: MATH 250 Student Number: Second Midterm Exam Instructor: March 29, 2007 Section: There are 6 multiple choice questions and 6 partial credit questions. In order to obtain full credit for the partial credit problems, all work must be shown. Credit will not be given for an answer not supported by work on a partial credit problem. THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINA- TION. For multiple choice problems, write the letter of your choice in the space provided below. Your Answer : Points awarded 1. (5 pts) — Q. 7 (10 pts) 2. (5 pts) — Q. 8 (15 pts) 3. (5 pts) — Q. 9 (15 pts) 4. (5 pts) — Q. 10 (10 pts) 5. (5 pts) — Q. 11 (10 pts) 6. (5 pts) — Q. 12 (10 pts) MATH 250 —Second Midterm Exam— 1. (5 points) Using Euler’s formula determine which of the following equalities is not true. a) 627” : 1. 62—21. : —62 cos 2 — 2'62 sin 2. U‘ 13m : _1 d 6 ( ( ( ) c) €2+2i : 62 cos 2 + 2'62 sin 2. ) 2. (5 points) Knowing the differential equation for a spring—mass system, which of the following statements is not true? (a) The equation describes a damped spring—mass system with a mass 4 and spring constant 1. (b) The system has no external forcing. (c) Irrespective of the initial conditions, the system will eventually come to rest. (d) The quasi frequency of the system is 1/2. 3. (5 points) Suppose you know that the solution of a certain mass—spring system is u(t) : 3e_t sin(7rt). Find quasi period and phase. (a) Quasi period is 2 and phase is 0. (b) Quasi period is 2 and phase is 7r/2. (c) Quasi period is 7? and phase is 0. ) (d Quasi period is 3 and phase is 0. Page 2 of 9 MATH 250 —Second Midterm Exam— 4. (5 points) A certain undamped mass—spring system is described by the following differ— ential equation. u” + 5u : cos wt. Find the value of w for which resonance occurs. (a) 5, (b) x/5. (c) ln 5. ) (d There is no such w. 5. (5 points) Suppose you know that the solution of a certain mass—spring system is u(t) : 2e_t/3 cos(5t — 6), where the phase 6 is unknown. What is the smallest time T for which this solution is guaranteed to satisfy < 1 for all t > T? 6. (5 points) Which of the following is the general solution of the following differential equation y” + y’ — 2y : —7+ 215+ 3cos3t — 36-2" — 11sin3t? (a) : 3 — t" 1.56—2t + sin 3t — 2 cos 315 + cue—t + c262t. (b) : 3 — t" 2t€_2t + 215 sin 315 + 3 cos 315 + cue—2’f + cget. (c) : 3 — t — e—2t + cos 3t — 2 sin 3t + cue—2" + cget, (d) y(t) : 3 — t" tie—2t + sin 315 + ole—2t + cget, Page 3 of 9 MATH 250 —Seeond Midterm Exam— 7. (10 points) (a) Use the method of reduction of order to find a second solution of the differential equation t2y” — 2ty’ + 2y : 0, 26> 0. knowing that y1(t) : t2 is a solution. (b) What is the general solution of the above equation? Page 4 of 9 MATH 250 —Second Midterm Exam— 8. (15 points) A mass of 0.25 kg is attached to a spring and causes it to stretch 12.25 cm. The mass is attached to a damper which exerts a force of 2 N when the velocity of the mass is 1 m / s. The mass is pulled down 4 cm from its equilibrium position and given an initial downward velocity of 16 cm / s. (a) Write an initial value problem describing the situation. (b) Solve the equation from part (a). (c) When is the first time to when the mass returns to its equilibrium position? Page 5 of 9 MATH 250 —Second Midterm Exam— 9. (15 points) Determine a suitable form for a particular solution Y(t) if the method of un— determined coePficients is to be used for each of the following nonhomogeneous equations. DO NOT determine the values of the coefficients. (a) y” + 2y’ + y : 3te_t + e_t cost (b) y" + 9y : —2t sin(3t) + 7cos(3t) + et sin(3t) (C) y” + 4y’ + 13y : 0.75t6_2t cos(3t) + 57 Page 6 of 9 MATH 250 —Second Midterm Exam— 10. (10 points) Solve the following initial value problem y” — mg + 36y : 0, y(0) : 1, y’(0) : 0- Page 7 of 9 MATH 250 —Second Midterm Exam— 11. (10 points) Solve the following initial value problem y” — 3y’ — 4y : 4 — 66% y(0) : 0, y’(0) : 0- Page 8 of 9 MATH 250 —Second Midterm Exam— 12. (10 points) Suppose f(t):(t2+l)e_3t. (a) Write the integral defining the Laplace transform of f(t). (b) For which values of s does the integral in part (a) converge? (c) Calculate the integral in part (a) explicitly. Page 9 of 9 ...
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midterm2_2007_spring - Name: MATH 250 Student Number:...

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