exam 1 2011

exam 1 2011 - pool During the cleanup effort in the...

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Math 222 EXAM I, September 21, 2011 Read each problem carefully. Show all your work for each problem! No Calculators! 1. (16 points) Consider the differential equation y' = ay + b, where a and b are constants. (a) Find a pair of values of a and b such that any solution yet) of the above equation approaches - 2 as t -+ +00. (b) Sketch the direction field of the equation, corresponding to the values in part (a). (e) Find the general solution of the equation, corresponding to the values in part (a). 2. (16 points) Solve the initial value problem y' + 2y = ie:", y(l) = o. Is the above equation linear? 3. (16 points) Solve the initial value problem y' = 2t(1 + y2), yeO) = O. What is the region of validity of the obtained solution? 4. (16 points) Solve the initial value problem 2ty' + 3y = t, 1 y(l) = 5' 5. (18 points) Following an accident at a nuclear plant, 10 5 m 3 of contaminated water containing 6 kg of radioactive isotope cesiurn-I37 was deposited in a containment
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Unformatted text preview: pool. During the cleanup effort in the aftermath of the accident, the contaminated water passes through a purifier at a rate of 10 4 m 3 jday, i.e., the contaminated water in the pool is replaced by clean water at this rate. How many days should the purification last for the level of the isotope to fall below the safe limit of 2 x 10-13 kg/m 3 ? (Hint: with good accuracy In3 ~ 1.1). 6. (18 points) Consider the initial value problem dy (y)2 dt = 1 + t ' yeO) = 1. (a) Use the Euler's method with step size h = 0.2 to find an approximate solution of this problem at t = 0.2,0.4,0.6. (b) Find an explicit solution of this initial value problem and compare it with the result in part (a). How well did the Euler's method approximate the solution at these values of t?...
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