MIT 18 03  Final Examination 18.03 (17 pages)
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Final Examination 18.03
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Exam
 Pages:
 17
 School:
 Massachusetts Institute of Technology
 Course:
 18 03  Differential Equations
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18 03 Final Examination 9 00 12 00 May 18 2010 Your Name Recitation Leader Recitation Time Do not turn the page until you are instructed to do so Write your name your recitation leader s name and the time of your recitation Show all your work on this exam booklet When a particular method is requested you must use it No calculators or notes may be used but there is a table of Laplace transforms and other information at the end of this exam booklet Point values out of a total of 360 are marked on the left margin The problems are numbered 1 through 10 1 2 3 4 5 6 7 8 9 10 1 a and b concern the tritium that is leaking from the Oyster Creek Nuclear Generating Station into the aquifer in New Jersey at a certain rate which we will assume is one kilogram per year The half life of tritium is 12 years 6 a Ignoring other e ects other sources or sinks of the tritium set up a di erential equation for the amount of tritium in the aquifer as a function of time For full credit determine any constants in the equation 6 b If this leak goes on for a long time what will the tritium load in the aquifer be How many kilograms Parts c g of this problem will concern the di erential equation 6 dy y2 x dx 4 c Let y x denote the solution to this equation such that y 1 0 Use Euler s method with step size 12 to estimate y 2 3 dy y2 x dx 4 d Sketch the isoclines for slopes m 1 m 0 and m 1 on the plane below 3 e On the same plane sketch the graph of the solution y x with y 1 0 6 f For this same solution suppose that y x achieves a minimum at x a What is y a in terms of a 6 g Estimate the value of y 100 1 Continuing with 2 In a c we consider the autonomous equation x x3 x2 2x 10 a On the vertical line below sketch the phase line of this equation 10 b Sketch the graphs of some solutions Be sure to include at least one solution with values in each interval above below and between the critical points 6 c Suppose x 0 is quite small say 0 1 For t 0 x t is best approximated by 0 1 eat for what value of a
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