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Unformatted text preview: Solutions to Exam 2: MAP 2302 * November 6, 2008 Name : Student ID : This is a closed book exam and the use of calculators is not allowed. 1. Let an amount x (in kg) of a radioactive substance decay with rate constant 1/hr (the decay rate of the substance is proportional with x ). This substance is entirely converted in a non radioactive substance with amount y , which in turn is degraded by its environment with a rate constant 2/hr (the degradation rate is proportional with y ). If initially we have that x (0) = 1kg and y (0) = 0kg, find x ( t ) and y ( t ) for all t . Solution : x = x, x (0) = 1 (1) y = x 2 y, y (0) = 0 (2) Solving the (linear, separable) first order equation (1) yields x ( t ) = c e t and c = 1 since x (0) = 1 = c e . Plugging this into equation (2), we see that that equation is linear: y + 2 y = e t . The integrating factor is ( t ) = e 2 t and thus d dt ( e 2 t y ) = e t ....
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 Summer '06
 DeLeenheer

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