MATH 241 Final: MATH 241 UPenn 241Fall 05makeup Final Exam
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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. your exam book and this question sheet but not your "cheat sheet". Part (ten points. each~ do four: find the. fourier series. for the. function which is. [x : use the fourier sine transform to solve the forrowingversion of the heat equation : 2 au = k a ~ rjt ax t > 0, x> 0 subject to u(o,t) = 0, u(x,o) = e~. Use the fact (easy to prove and ~imil~r to t. ::ipl~r. etransform. 1+ cr: 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: solvethe (boundaryfess) wave equation ~.
