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Page 1 of 16Student Name:Student Net ID:UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDEPARTMENT OF MATHEMATICSMATH 286 SECTION X1 – Introduction to Differential Equations PlusFINAL EXAMINATIONDECEMBER 17, 2013INSTRUCTOR: M. BRANNANINSTRUCTIONSThis exam is three (3) hours long. No personal aids or calculators are permitted.Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.GOOD LUCK!PLEASE NOTE:“Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Question:1234567TotalPoints:142599141514100Score:
Student Net ID:MATH 286 X1Page 2 of 16SOME USEFUL FORMULAS:eB=Xk=01k!Bk=I+B+12!B2+13!B3+. . .x(t) = Φ(t)Φ(a)-1x(a) + Φ(t)ZtaΦ(s)-1f(s)dsabcd-1=1ad-bcd-b-caan=1LZL-Lf(t) cosnπtLdt,bn=1LZL-Lf(t) sinnπtLdt1. Letf(t) be theπ-periodic function defined byf(t) =(1-π2< t <0,t0tπ2.(a) (2 points) Sketch the graph offover a few periods.(b) (6 points) Leta02+L) be the Fourier series forf. Calculatethe Fourier coefficientsa0, anandbn(n1).
Student Net ID:MATH 286 X1Page 3 of 16
Student Net ID:MATH 286 X1Page 4 of 16(c) (2 points) Does the Fourier series forfconverge tof(t) at every pointt? Whatdoes the Fourier series converge to whent= 0?

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