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Exam1005F2008wAnswers - CARLETON UNIVERSITY FINAL...

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CARLETON UNIVERSITY FINAL EXAMINATION December 2008 DURATION: 3 HOURS Department Name and Course Number: Mathematics and Statistics, MATH 1005F Course Instructor: Dr. A. Alaca PART I: Multiple Choice Questions. No partial marks. Circle the correct an- swer. [3] 1. What is the orthogonal trajectories of the family of curves y = ke x ? (a) y 2 ¡ x = c (b) y 2 + 2 x = c (c) y + x 2 = c (d) y + 2 x 2 = c (e) y 2 + x 2 = c [3] 2. If y is the solution of the initial value problem y 0 = 2 xy; y (0) = 1 ; what is y (1) ? (a) e (b) e ¡ 1 (c) ¡ e (d) e 2 (e) 2 e [3] 3. Let y 00 ¡ 6 y 0 + 8 y = e 4 x . Which of the following is an appropriate form of a particular solution y p when using the method of undetermined coe±cients? (a) y p = Ae 4 x (b) y p = Axe 4 x (c) y p = Ax 2 e 4 x (d) y p = ( Ax + B ) e 4 x (e) y p = Ae 4 x + Be ¡ 4 x [3] 4. What is the general solution of x 2 y 00 + 5 xy 0 + 5 y = 0, x > 0? (a) y = x ¡ 1 h c 1 cos(ln x ) + c 2 sin(ln x ) i (b) y = x ¡ 1 h c 1 cos(2 ln x ) + c 2 sin(2 ln x ) i (c) y = x ¡ 2 h c 1 cos(ln x ) + c 2 sin(ln x ) i (d) y = e ¡ 2 x h c 1 cos( x ) + c 2 sin( x ) i (e) y = e ¡ x h c 1 cos(ln x ) + c 2 sin(ln x ) i
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[3] 5. If y
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