1b-2009-exam_3_practice_solutions

1b-2009-exam_3_practice_solutions - Math 1B Exam #3...

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Unformatted text preview: Math 1B Exam #3 Practice Solutions Fall Program for Freshmen 2009 Fred Bourgoin 1. Use Eulers method with step size 0.1 to estimate y (0 . 3), where y is the solution to the initial-value problem y = x + xy , y (0) = 1. Solution. It is easier to organize things in a table. x y ( x ) y ( x ) y (0) = 0 y (0) = 1 0.1 y (0 . 1) 1 + (0 . 1)(0) = 1 y (0 . 1) . 2 0.2 y (0 . 2) 1 + (0 . 1)(0 . 2) = 1 . 02 y (0 . 2) . 404 0.3 y (0 . 3) 1 . 02 + (0 . 1)(0 . 404) = 1 . 0604 2. A tank contains 20 kg of salt dissolved in 5000 L of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour? Solution. If we let y ( t ) be the amount of salt in the tank at time t (in minutes), the problem can be translated as the differential equation dy dt = [rate in]- [rate out] = (0 . 03)(25)- y 5000 25 = 0 . 75- . 005 y with initial value y (0) = 20. The equation is separable. dy dt = 0 . 75- . 005 y = - 200 dy dt = y- 150 = dy y- 150 =- dt 200 = Z dy y- 150 = Z- dt 200 = ln | y- 150 | =- t 200 + C = | y- 150 | = Ce- t 200 = y- 150 = Ce- t 200 = y = 150 + Ce- t 200 Now plug in the initial condition: y (0) = 150 + C = 20 = C =- 130 ....
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1b-2009-exam_3_practice_solutions - Math 1B Exam #3...

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