prelim_2_solutions - Math 2930 Prelim II, 7:30~9prn,...

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Unformatted text preview: Math 2930 Prelim II, 7:30~9prn, November, 2010 Answer each of the following 6 questions (20 pts. each) —-— show all work (closed book, 1 “cheat sheet”, no calculators). Academic integrity on the part of each student is presumed — Violations will be dealt with swiftly and justly. I. Consider the ODE 56 ~§~ 22E: + x 2 4cos(2r): (a) Find the general solution. (b) What is the steady—state solution (for t > 0 “large”). d2)» dy dy 2. SolvetheIVP fi—-+r——4 =0, r>0, 1 :1, mm 1:0. dfi dz y y” dt() 3. Consider the ODE y’” + 4 y' = fix). For each case, give the general form of the particular solution using the method of undetermined coefficients. Do not evaluate the coefficients. (a)f(x) = 5 + 8x3, (b)f(x) = xsin(5x) (c)f(x) = cos(2x), (d)f(x) = 23in2 x. 4. Consider the EV]? y" +16y = O, y(0) m y(L) = 0. (a) Find all values of L > 0 such that the BVP has a nontrivial solution (b) Determine the nontrivial solution corresponding to each of the values of L found in (a). (c) For what values of L > 0 does the BVP admit a unique solution — what is that solution? 5. Each of the following Fourier series converge to the functionsf(x) 2 x on the indicated intervals. 00 I144 co ~i . if 4 cos nx x222( srn(nx), —7r<x<7r,andx=———— Z (2 ),O$x<n. W1 n 2 71' n=l(odd) 1’7 (a) In each case, to what value does the series converge when (1')): = "*7? f3, (ii) x = 71*, (iii) x = I In + 7: f2. ? (6 answers required in total) (b) What is the Fourier series of [x ,myr < x < 7: (period 27:)? 6. Consider the IBVP u! :ufl, O<x<7t, r>0, ux(0,t) 2“: ux(7r,r) = 0, u(x,0) = 5 ~+~ 3x. Obtain the steady-state solution lim n(x,r). Justify your answer. 1am 3"”‘5figfifi . “gr: W” fl???“ é} MW r3?» 1%» w k3,.» M I #55:) I fig 111:, - m ‘“‘ =3 : 5.2,? 5:??qu jg; _rt"" 1mm 511W MW” .e ‘p I} «:7; a, M» w.“me m.“ . was; .5“, ' -* 6‘4er ‘ WV 95: a my ‘ 3 . Wmmmmm wg-m. . WW3 Nfimm. _ 51% 3:: y if? win m. ...
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prelim_2_solutions - Math 2930 Prelim II, 7:30~9prn,...

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