M2solns

# M2solns - MATH 405 Midterm 2 Friday NAME D LMT/om g This...

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Unformatted text preview: MATH 405: Midterm 2 Friday, November 17, 2006 NAME: D LMT/om g This exam consists of 5 problems, with points as labelled. Please write each of your answers clearly and legibly. Don’t forget to explain yourself, and show all of your work! There are no calculators, cellphones, iPods, etc permitted during the exam. 1. (20 pts) The ODE for the position \$(t) of a mass m attached to a spring (with constant [9) including damping (coefﬁcient 6) is: mac” + bx’ + km = 0 (a) Show that this is a Sturm-Liouville equation. (b) State the orthogonality relation between the eigenfunctions acn of the equation, on some time interval (0, T). What sorts of (boundary) conditions on 55(2?) would this require? (Ifng (; gTuRM’LIouVII/Lﬁ stE/‘w/ (1/,777c P266” [#49 lE; 6W” ll:’% 7/ / v [03 [Xmﬂle/tlfWWQ/zt “)0 “714’”1 BC O ‘ ﬂA/‘(Tl-HNG 0’: \$1341: Fokh /— kX/OB *AX’ﬂéBi‘ 0 K‘s/f A/‘li/VKONSWNTEA CE? 2. (20 pts) Laplace’s equation V2q§ = 0 in cylindrical coordinates for ¢ = (1)0", (9) can be written as 8% 18¢ 18% W+FE+FEEFO Using separation of variables ﬁnd the ODE for the radial dependence, and solve it. 7% 93 : Km 14/9) i5 HR” +—g—HR’+}‘—a— CEH” :- D U3— Hit % FDR/(+rR/3fg/j1r—k R 7%" H Wave; ——Falé SPEC?“ 6455 [5:6 D / Dam Bat-E {2007‘ #19135 (— 1.: 40 IS INéLubEb #5185 % MOWTN\$ @ 3. (20 pts) Find the general solution to the following ODE for y” + 67r2y = 7‘(a:) given that the nonhomogeneous function 7‘ is deﬁned by its Fourier series: 00 371'2 , 7°(x) = 2 mg + 14 s1n(n7rac) n=0 Note you can leave the particular solution as an inﬁnite sum! “bib .9 ' (( + gr? : 2772 V . ’—> 7/) I 7.4 Mék/Lf g)” (/1545 r’ﬂ L 9x4 M B V ‘ N _ 74 4 ( 5/0”} {34 / w “Marriage; 9'56 771%?" ﬁg :[email protected] @ 4. (20 pts) For the given function f = sin(a:) on the domain (—1, 1), consider a Legen— dre expansion f = ZanPn(:r). Write out the relation for the coefﬁcients (see last page), and calculate explicitly a0 and a1. Are either of them zero? «ED/RE THIS EXPANSION 5. (20 pts) Find the general solution for the following ODE for xzy”—(a:+2)y=0 é:/ 1X7 M "\$qu Ff?» EEK/i «45 ﬁber "4 W/r‘: M _, @ I‘M ﬁr) a“ xfyf/r ——/7:s+r [/aw’EST’ Pa~J€R ﬁx? TERMS : Qb'f O r/~(\ Q0 4% “>6150 % 9- r- =0 > “’3” \$24 MTE 4'“:- u \T H V \U ( ...
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