math2930_2011fa_prelim1.pdf - Math 2930 Fall 2011 Prelim...

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Math 2930, Fall 2011, Prelim Exam 1 Thursday Sept. 29, 7:30-9:00pm This exam is closed book and closed notes. No calculators. Please turn off all cell phones and other communication devices. Academic integrity is expected of all students at all time whether in the presence of absence of members of the teaching staff. If you understand this and agree that you will neither give, nor use, nor receive unauthorized aid in this examination, please sign the Academic Integrity space on the examination booklet. Thank you. 1. [15 points] Suppose that y ( t ) satisfies the differential equation dy dt = sin y, with initial condition y (0) = π/ 4. Which of the following is the limit of y ( t ) as t → ∞ ? (a) π/ 2, (b) - π/ 2, (c) 0, (d) π , (e) 2 π . 2. [15 points] Test the equation e y dx + ( xe y + 2 y ) dy = 0 for exactness, and solve it if it is exact. 3. [15 points] Solve the differential equation x dy dx - 3 y = x 4 . 4. [15 points] Use the Euler method to find the approximate solution to dy dt = - yt 2 , with initial condition y (0) = 20 and step size of ∆ t = h = 1. Calculate the approximate
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