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1005 test 1 - MATH 1005F Solutions to Test 1 October 5 2009...

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MATH 1005F Solutions to Test 1 October 5, 2009 1 This test paper has two parts and total of 30 marks. Part I has 3 multiple choice questions. Part II has 4 long answer questions. It cannot be taken from the examination room. Only nonprogrammable calculators are allowed. Duration: 50 Minutes. NAME : STUDENT NO : PART I: Multiple Choice Questions. No partial marks. Circle the correct answer. [2] 1. If y is the solution of the initial value problem y + 2 y = e - x , y (0) = 1, what is y (2) ? (a) 2 e (b) e (c) e - 1 (d) e - 2 (e) e 2 [2] 2. If y is the solution of the initial value problem y = x y , y (0) = 2, what is y (1) ? (a) 5 (b) - 5 (c) 5 (d) - 5 (e) 4 [2] 3. What is the orthogonal trajectories of the family of curves y = ke 6 x ? (a) y 3 + x 3 = c (b) y 3 + x = c (c) 3 y 2 + x = c (d) 6 y +3 x 2 = c (e) 6 y 2 + e - 6 x = c PART II: Long answer questions. Show all your work. [6] 4. Find the general solution of the differential equation y = x 3 + y 3 xy 2 . Solution: y = x 3 + y 3 xy 2 = 1 + v 3 v 2 , y = xv y = v + xv v + xv = 1 + v 3 v 2 = v 2 dv = dx x v 3 3 = ln | x | + ln c, c > 0 y 3 = 3 x 3 ln( c | x | ) .
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MATH 1005F Solutions to Test 1 October 5, 2009 2 [6] 5.
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