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Unformatted text preview: MATH 251  Ans Key Examination II November 4, 2008 Name: I Newton Student Number: 2 Section: 12 This exam has 13 questions for a total of 100 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. You may not use a calculator on this exam. Please turn off and put away your cell phone. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: Total: Do not write in this box. MATH 251 EXAMINATION II  ANS KEY November 4, 2008 1. (5 points) Match the sketches of phase portraits for 2x2 homogeneous linear systems x ′ = A x with the names of their critical points at the origin. 1 2 3 4 5 6 (a) saddle 5 (b) node 1 (c) proper node 4 (d) center 2 (e) spiral 3 Page 2 of 10 MATH 251 EXAMINATION II  ANS KEY November 4, 2008 2. (5 points) Match the following formulas for general solutions of 2x2 homogeneous linear systems x ′ = A x with the sketches of the phase portraits given in Problem 1: (a) c 1 e 2 t parenleftbigg 1 1 parenrightbigg + c 2 e − t parenleftbigg 1 1 parenrightbigg 5 (b) c 1 e t parenleftbigg cos t 2sin t parenrightbigg + c 2 e t parenleftbigg sin t 2cos t parenrightbigg 3 (c) c 1 e − t parenleftbigg 1 parenrightbigg + c 2 e − t parenleftbigg 1 parenrightbigg 4 (d) c 1 e t parenleftbigg 1 parenrightbigg + c 2 e − t parenleftbigg 1 parenrightbigg 5 (e) c 1 parenleftbigg cos t 2sin t parenrightbigg + c 2 parenleftbigg sin t 2cos t parenrightbigg 2 3. (5 points) Match the three adjectives for the critical point parenleftbigg parenrightbigg of a 2 × 2 homogeneous linear systems x ′ = A x with the five general solutions given in the table below by placing one of the letters A , U , or S in each of the five blanks. (a) c 1 e 2 t parenleftbigg 1 1 parenrightbigg + c 2 e − t parenleftbigg 1 1 parenrightbigg U Use (b) c 1 e t parenleftbigg cos t 2sin t parenrightbigg + c 2 e t parenleftbigg sin t 2cos t parenrightbigg U A for asymptotically stable (c) c 1 e − t parenleftbigg 1 parenrightbigg + c 2 e − t parenleftbigg 1 parenrightbigg A U for unstable (d) c 1 e t parenleftbigg 1 parenrightbigg + c 2 e − t parenleftbigg 1 parenrightbigg U S for stable (e) c 1 parenleftbigg cos t 2sin t parenrightbigg + c 2 parenleftbigg sin t 2cos t parenrightbigg S Page 3 of 10 MATH 251 EXAMINATION II  ANS KEY November 4, 2008 4. (5 points) When an object with mass 5 kg is attached to a spring, the object stretches the spring by 2 m....
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations, Cos, Constant of integration, Boundary value problem, Picard–Lindelöf theorem, ANS KEY

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