02spex3 - 5 (4) Find the general solution of d 2 y dt 2 + 2...

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MATH 222 — THIRD MIDTERM April 10, 2002, 9:55am–10:45am Your Name: Your TA: (circle one) Chris Alfeld Graham Jonaitis Andy Raich Joshua Rushton Fernando Miranda Score 1: 2: 3 or 6: 4: 5: Total: THIS EXAM HAS SIX PROBLEMS YOU SHOULD ONLY DO FIVE OF THESE EVERYONE MUST DO PROBLEMS 1,2,4,5 YOU MUST CHOOSE BETWEEN PROBLEMS 3 AND 6 DO EITHER 3 OR 6 BUT NOT BOTH 1
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2 (1) Find the solution of the di±erential equation dy dx = (1 + y 2 ) sin x, which satis²es y ( π 2 ) = 3.
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3 (2) Find the solution of the di±erential equation x dy dx + (1 - x ) y = 1 - x which satis²es y (1) = 0.
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4 Do either this problem or problem 6 (on polar coordi- nates) but not both (3) In the problems on this page x is an abitrary real number. (a) Compute z = x + i 7 + i (i.e. write z as a + bi with a and b real.) (b) Assuming x > 0, draw x + ix 3 and compute arg( x + ix 3) (c) Assume 0 < x < π/ 2. Draw e ix , e 2 ix and e ix + e 2 ix in one fgure. Then compute z = e ix + e 2 ix , i.e. rewrite z = a + bi with a and b real. [Continue on reverse side]
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Unformatted text preview: 5 (4) Find the general solution of d 2 y dt 2 + 2 dy dt + y = 0 , where is some positive constant. Your answer should be valid for all possible choices of &gt; 0. 6 (5) Find the general solution of d 2 y dt 2 + 3 dy dt + 2 y = cos t. 7 Do either this problem or problem 3 (on complex num-bers) but not both (6) (a) Find the equation in Cartesian Coordinates for the curve which in Polar Coordinates is given by r 2 = sin cos . (b) Find the equation in Polar Coordinates for the curve which in Cartesian Coordinates is given by x 4 + y 2 = 3 x . (c) Consider a Logarithmic Spiral which in Polar Coordinates is given by r = e / 2 . For which value(s) of between 0 and 2 does this spiral have a horizontal tangent? [Continue on reverse side, if necessary]...
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02spex3 - 5 (4) Find the general solution of d 2 y dt 2 + 2...

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