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7 be careful to avoid making grievous errors that are

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(7) Be careful to avoid making grievous errors that are subject to heavy penalties. (8) If you need more blank paper, ask a proctor. Out of fairness to others, please stop working and close the exam as soon as the time is called. A significant number of points will be taken off your exam score if you continue working after the time is called. You will be given a two-minute warning before the end. 1 2 3 4 5 6 7 8 9 10 11 12 Total 1
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1. (20 points) Solve the differential equation: a) (10 points) x 3 e y - y 0 = e x + y . b) (10 points) yy 0 = cos x, y (0) = 9 . 2
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2. (20 points) Solve the differential equation (Find the general solution): a) (10 points) y 00 + 7 y 0 = 0. b) (10 points) y 00 + 6 y 0 + 10 y = 0. 3
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3. (10 points) Solve the following initial value problem: y 00 + 10 y 0 + 25 = 0 , y (0) = 3 , y 0 (0) = 0 . 4
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4. (10 points) A radioactive sample decayed to 80% of its original mass in 1 year. Find the half-life time for this sample. 5
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5. (10 points) A cup of iced coffee with temperature 10 C is placed in a room with temperature 25 C . After 10 minutes the temperature of the cup was equal to 15 C . Find the temperature of the cup after an hour. 6
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Test the following series for convergence: 6. (10 points) X n =1 3 n n + 3 7. (10 points) X n =1 n + 5 n - 10 7
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8. (10 points) X n =1 sin n - 1 n 3 + 1 9. (10 points) X n =2 ( - 1) n ln n 8
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10. (15 points) Consider the series X n =1 ( - 1) n ( x + 1) n n · 7 n . a) (10 points) Find the radius of convergence b) (5 points) Find the interval of convergence 9
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11. a) (10 points) Find the Taylor series for the function f ( x ) = 1 (1+ x ) 2 with center at a = 5. b) (5 points) Find the radius of convergence for this series 10
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12. (10 points) Present the integral as a sum of infinite series and estimate it with error within 0 . 001: Z 0 . 1 0 sin x - x x 3 dt. 11
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