m251_problem_set1[fa10]

m251_problem_set1[fa10] - MATH 251.6/8/11/12 Problem set #1...

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MATH 251.6/8/11/12 Problem set #1 Due: 9-30-2010 1. D’oh! Yet again Homer Simpson has had one Duff beer too many before work. As a result, water containing 5 kg/m 3 of radioactive plutonium-244 waste is flowing into Springfield’s reservoir at a rate of 4 m 3 per minute. The reservoir initially contains 2000 m 3 of fresh water. The mixture (assuming uniform concentration) is drawn off at a rate of 2 m 3 per minute. Find an expression for the concentration of radioactive material in the reservoir at time t . 2. The process of radioactive decay is described by the equation y ′ = – r y , where r is a positive constant (the decay constant of the radioactive material). Given that Plutonium- 244 has a half-life of 8 × 10 7 years, find its decay constant r by first solving the initial value problem: y ′ = – r y , y (0) = y 0 (where y 0 > 0). Then use the half- life to find r . 3. Consider the autonomous equation y ′ = y 3 ( y – 1) 2 (3 y ± 24). (a) Find all equilibrium solution(s), and classify the stability of each equilibrium solution.
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m251_problem_set1[fa10] - MATH 251.6/8/11/12 Problem set #1...

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