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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.
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 Spring '07
 TERRELL,R

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