M251final(sp09) - MATH 251 FINAL EXAMINATION May 6, 2009...

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Unformatted text preview: MATH 251 FINAL EXAMINATION May 6, 2009 Name: Student Number: Section: This exam has 18 questions for a total of 150 points. 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. The point value for each question is in parentheses to the right of the question number. A list of Laplace transforms is attached as the last page of this booklet. It can be removed for easy reference during the examination. You may not use a calculator on this exam. Please turn off and put away your cell phone. 1 : 2-14: 15: 16: 17: 18: Total: Do not write in this box. MATH 251 FINAL EXAMINATION May 6, 2009 1. (7 points) (a) (4 points) Match each of the sketches, lettered A through D, depicting the graphs of certain mass-spring systems given by the equation mu + u + ku = F ( t ), with the most suitable description from the list below. 1 0.5 0.5 1 2 4 6 8 10 t 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 2 4 6 8 10 t (A) (B) 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 2 4 6 8 10 t 8 6 4 2 2 4 6 8 2 4 6 8 10 t (C) (D) underdamped free vibration : resonance : overdamped free vibration : undamped free vibration : (b) (3 points) Which graph best represents a typical nonzero solution of a mass-spring system described by the equation y + 25 y = 0? Page 2 of 13 MATH 251 FINAL EXAMINATION May 6, 2009 2. (6 points) Consider the first order linear equation ty - 2 y = e t sin2 t, t > . What is a suitable integrating factor that could be used to solve this equation? (a) ( t ) = t 2 (b) ( t ) = e 2 t (c) ( t ) = e 2 t (d) ( t ) = 1 t 2 3. (6 points) Given that the differential equation below is an exact equation. Find the value of . (6 x 2 y 2 + y e xy ) + ( x 3 y + xe xy + 1) y = 0 (a) 0 (b) 3 (c) 4 (d) 12 Page 3 of 13 MATH 251 FINAL EXAMINATION May 6, 2009 4. (6 points) The autonomous differential equation y = 6 y- y 2 has two equilibrium solutions that are (a) asymptotically stable at both y = 0 and y = 6....
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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M251final(sp09) - MATH 251 FINAL EXAMINATION May 6, 2009...

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