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**Unformatted text preview: **18.03 Final Examination 1:30–4:30, May 23, 2006 Your Name Recitation Leader Recitiation 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 the exam booklets. 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. Recitation Leaders: James Albrecht, Peter Buchak, John Bush, Denis Chebikin, Sunhi Choi, Craig Desjardins, Ching-Hwa Eu, Chuy- ing Fang, Pak Wing Fok, Austin Ford, John Francis, Matthew Gelvin, Teena Gerhardt, Shan- Yuan Ho, Sabri Kilic, Boguk Kim, Wyman Li, William Lopes, Anjana Mohan, Jean-Christophe Nave, Josh Nichols-Barrer, Olga Plamenevskaya, Pavlo Pylyavskyy, Charles Rezk, Ruben Rosales, Yanir Rubinstein, Jake Solomon, Jeff Viaclovsky, Fangyun Yang 1 2 3 4 5 6 7 8 9 10 1. Parts (a) and (b) are about the Symbionese Liberation Bank, which offers a bank account paying an interest rate I which depends upon the amount of money x ( t ) in the bank account (in a way that is constant through time): I = I ( x ), for a non-constant function I ( x ). [6] (a) Write down a differential equation for x ( t ), if my rate of savings is given by q ( t ). [4] (b) Is this differential equation linear? Parts (c) and (d) deal with Euler’s method applied to the ODE y = 1 + xy . [6] (c) Estimate y (1) using Euler’s method with stepsize 1 / 2, where y is the solution with y (0) = 1. [4] (d) Is the Euler estimate too high or too low? 1 1. (continued) In (e) – (g) we consider the autonomous equation ˙ x = x 3 − x . [5] (e) Sketch the phase line in the space below. [5] (f ) 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] (g) Suppose that x ( t ) is a nonconstant solution to ˙ x = x 3 − x such that x 0. If x (2) = 0, ¨ what is x (2)?...

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