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241F05makeup-final - Math 241 Fan 05 Final Exam B...

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Math 241 Fan 05 Final Exam B Instructions:. Do four of five in Part One and six of the seven in Part Two. If you try more than the required number, only your best will be counted. You may use one standard size 8.5 by 11 inch "cheat sheet" and your calculator. Turn in your exam book and this question sheet but not your "cheat sheet". Part ~ (Ten points. each~ Do four} 1. Find the.Fourier series.for the.function which is. 2.3tperiodic and whose.value.for o ~ x < 27tis [x [ ---------------------------------------------- 2. Use the Fourier sine transformto solve the forrowingversionof the heat equation : 2 au = k a ~ t > 0, x> 0 subject to u(o,t) = 0, u(x,o) = e~. Use the fact (easy to rJt ax prove and ~imil~r to T.::Ipl~r.etransform. ).that fJ [ e -x J = ~ /I. 1+cr 3. Solve the Sturm - Liouville problem y" -f Jr.y = 0- subject to y'(o)- = 0- and Y'(3r} = 0- on the interval [0, 3tJ. Be sure to deliver ALL eigenvalues and ALL eigenfunctions and to demonstrate. that yo.uhave.found ALL such ------------------------------------------------ 2 2 4. Solvethe (boundaryfess) waveequation ~.; = c2 ~; -00 < x < 00, t> 0 subject tt>u(x,o}= J(xHiMialdeflecOOfi}and ~ (x, 0}= o.(inifl.al speedzero-}. FOfbestrestllts
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