2930Prelim 2 Fall 2009_noSolution.pdf

# 2930Prelim 2 Fall 2009_noSolution.pdf - Mathematics 2930...

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Mathematics 2930. Fall 2009 Prelim I I Thursday, November 19, 2009 7.30 PM - 9 PM Instructions: Write your name, your TA’s name and section number on your exam booklet No calculators, other electronic devices, or notes are allowed in this exam. Give reasons for your answers and show all work There are 5 problems, each worth 20 points. Good luck ! 1. (a) (10 points) Find a general solution of: y - y - 2 y + 2 y = 0 (1) Here, y is a function of x for all real x and denotes a derivative with respect to x . Make sure that the correct number of arbitrary constants appear in your solution. Hint : The fourth degree polynomial equation: r 4 - r 2 - 2 r + 2 = 0 (2) has two roots r 1 = r 2 = 1. First find the other two roots r 3 and r 4 . (b) (10 points) Now consider the equation: y - y - 2 y + 2 y = e x (3) Find a particular solution of (3). 2. (20 points) Determine the unique solution x ( t ) of the following problem. ¨ x + 4 x = sin( t ) cos( t ) , x (0) = ˙ x (0) = 0 (4) where ˙ x dx/dt .

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