Practice Exam 1

Practice Exam 1 - 92.236 Engineering Dierential Equations...

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92.236 Engineering Diferential Equations Practice Exam # 1 Solutions Fall 2008 Problem 1. (10 points) Is y ( x ) = 1 /x is a solution of the d.e. y p = x - y 2 ? Why or why not? Left side of d.e: y ( x ) = 1 x = x - 1 y p = - x - 2 = - 1 x 2 . 3 pts. Right side of d.e: y ( x ) = 1 x x - y 2 = x - p 1 x P 2 = x - 1 x 2 . 3 pts. Left side n = right side, so y ( x ) = 1 x is not a solution of the d.e. y p = x - y 2 4 pts. Problem 2. (15 points) Initially there are 100 grams of a radioactive substance in a container. After 1 day, there are 90 grams left. a. How much of the substance will be left after 5 days? Let t denote time (in days) and let N denote the amount of radioactive substance (in grams) in the container at time t . Then N = N 0 e - kt where N 0 is the initial amount of the substance. We are told that N 0 = 100, so N = 100 e - kt . 3 pts. We are told that N (1) = 90, so 90 = 100 e - k · 1 90 / 100 = e - k ln(0 . 9) = ln ± e - k ² = - k k = - ln(0 . 9). 4 pts. Therefore, N (5) = 100 e - k · 5

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This note was uploaded on 11/01/2008 for the course MATH 236 taught by Professor White during the Fall '06 term at UMass Lowell.

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Practice Exam 1 - 92.236 Engineering Dierential Equations...

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