set2_prolemsandsolutions

set2_prolemsandsolutions - MA 527 EXAM 2 Fall 2010 Page 1/5...

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Unformatted text preview: MA 527 , ,, EXAM 2 Fall 2010 Page 1/5 t / V (10 pts.) 1. Assume that y(t) is a solution to the problem 2y” + 33/ + y = 6(t — with y(0) = 5 and y’(0) = 7. s 2) Find the Laplace Transform Y of y(t). (DO NOT FIND y(t). Just find Y(s).) ,. ~ + f _ ~35 :9 » a afsaf~597jt3£3 ) fl [Os +37 +635 Y 1 $437+ 35 4* l (20 pts.) 2. Graph the function f (t) given below. Next, express f (t) in terms of step functions and simple functions of t. Finally, compute the Laplace transform of f. 0 forOStSl f(t)= (t—1)for1£tg2 1 for2£t l ’A » . a i 9 41(45): CH (VD -“ ué€-3)j(e~i) + “(way 1 ~:_ 054‘)“ (MW -- Ct "52)“ [t ’3) ‘71:.» M“) Page 2/5 (20 pts.) 3. Find the Inverse Laplace Transforms of the following functions. L _ A g _ a _ (“2) a) 8 s “ ‘1’“ t w $+3»A(5H)r3 b) (affix +1) 5% 6H 2 (MB): + (M233 (8+1)2 3—3 C)(S+1)2+4 A'f'lzz ’l Cu) "5 LI Anny} Cb) “l "" * '7 ' “7:11 éfa 54' (Q’M/ Mga ’ 2””:66’33 Lleat (a), [4‘4 fl “~95 , ~ 1 5 19) j atlas“? fluvsna'sfl a H) Ht) [2(9) . 4&5) 60 925,4 :3 (t’9>€ MOVIE) c) g»; 3., [Gm—I] L{ H (M) :2 fl ’- 1 2 2 m2 8H)“ + 21 (5+1) +QA (EMU :2 (6H) #3 h M 2 a (Q ‘ (9+!)J+J (6H) +9 w t “b as 225 Page 3/5 (20 pts.) 4. Find all positive eigenvalues for the Sturm~LiouVille problem 3;” + Ay = 0 with y'(0) = 0 and y(2) = 0. (Don’t bother checking the A = 0 or A < 0 cases, and don’t find the eigenfunctions.) 2 5’ fl a+ ' 9c COSMK'tC {Eh/MK wimp} YUM/{’0} wh/w x5) . «4 mm 4" Z , "(0) ‘1 Cog/Aviafl, 40 C02: (7 am! y: chos/ozx I; [want g(2) 7-: s 0 o mad (099/430 9": “CM/162mm mm (bi/4% 5% {X Newt auz’w’r, WWW f“ m” a #4 a! / Map/[)Q/ rr« 1/)” a 2“ ’Z’ A) A 2W, PDSPHV/fi Qty/at/H/fll/dé‘) $3M » (34‘ .2 / M/fljil / a 2 1 2,—‘9- 1N1 (:7 511 gll [If a /) “fl ) W 2 “’“2 [6: fly /(1 /C4 (10 pts.) 5. Suppose 2:021 bn sin ms is the Fourier series for the function which is —$2 for —7r < m < 0 and is m2 for 0 g a: < 7r. Find 220:1 bfl. Note: You do not need to compute the bn to do thiswproblem. I; 1 v 94 I‘ J ' 9 ’1 a! 91 A L x ?arse\/mi55 3% + £04m Ham).. g: b” ’17 MM) M ’1“ “’I 'l‘ ' h’ “u even Oéiadaiaa ‘2 ’2‘] Q J T; Li .2 7(5 '17 QTL’ 4 77‘ £(«>M:;§ a auxin] : _____ 0 ll 0 ll 5 0 B m f. (10 pts.) 6. Show that the functions 1, 9:, and cosac are orthogonal on the interval —7r < cc < 71' using properties of integrals of odd and even functions when applicable. Given that f(x) = 01-1 +0250 +03 cosx, express 02 in terms of integrals over the interval —7r < a: < 7r involving f and the given functions. (if (v 4 S“t.x 4W0 ngnxmo ~ 7 ,v '1‘ “u '1‘ i "‘ by“ even eVeVl 0 W W W M ‘1" LI _, /\ (7 '7? W W‘X X’C’SX axlem : C ’f C . ,., u A L( 1 I C é(7 ‘ ’ ‘1? WWW “(’0 1‘ 91f "X “W 5 “CWT .w «(a :E’W‘r/ Sty QTY/3 (10 pts.) 7. Let Page 5/5 woo ‘ 0061,0407 -7( -wa ~ L a M e e W' (Y W O . I ("pan/)X B a . 'f -3 ,au/ ‘8 +11% (—1333 4 e C 6% T '\ ‘ a0 lei.” : 1:5?” beam/ea“ ...
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This note was uploaded on 04/03/2012 for the course MA 527 taught by Professor Weitsman during the Spring '08 term at Purdue.

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set2_prolemsandsolutions - MA 527 EXAM 2 Fall 2010 Page 1/5...

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