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test2AB_ans - TEST 2 A NO CALCULATORS MAP 2302 Name 5 0L U...

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Unformatted text preview: TEST 2 A (08/05/11) NO CALCULATORS ? ' MAP 2302 Name: 5 0L U T/OA/S Summer B 2011 3rd period Section 4787 Please use the blank space and the other side of the Sheet to write your solutions, clearly numbering them. 1.(2 pts) Find the Laplace transform of the function defined by f(t)=0, 0<tgi, f(t)=1 1<t§2 T 9% a 2/\ x: - and periodic with period T = 2. 021/“ {I} {/5 3 : gf (1L ,1 ‘¢ M jar tit/1‘s flaw/m. 7:2 536 u f" 153 “T” ’ 2 ”W9 7f“? “3‘211 43/3/3639 2‘12 pts) Find first two non—zero terms of the Taylor polynomial approximation (centered at $0 2 0) of the solution of the following initial value problem (write the corresponding Taylor polynomial): y' + 3tan(.7;y) : O, y(0) = 2 (/2) ‘ 'V” / :Ui) : A3€m(0) :0 V ”3720’; “3&68C’t13) (3 +203 > "3’ 3%.» $312 a a 1 Ma Ma: .. WEI/1‘ *1. " U 3. (2 pts) Find the recurrence relation for the coefficients in the power series expansion about .TQ— — O of a solution of the following equation. 010 ”a (1L ’ <30 WW .._— 1 AL ‘ & fx+i Law: K +1: aefix’ 7:1,“ 2;? KM; Zita/C y"+(:z7+1)y+4y=0 x ”’31-: 1 M» Mt? 1 E z; a a ’" d 12 3:1,— ,x» 1'; 13 C i . \ ii ’ :42 ‘7 111+ said M 12+; 4. (a)3(2 pts) In the method of variation of the parameter a particular solution of the non—homogeneous equation has the form yp(t) : 111 (t)y1(t) +v2(t)y2(t), Where y1(t) and y2(t) are linearly independent solutions of the corresponding homogeneous equation. Find y1(t) and y2(t) for the following equation, and set up the equations for v’1(t) and 0.30:). y" + y : sec3t (b)(2 pts) Solving the equations in part (a), find the particular solution of the given equation 5. (a)(2 pts) Find the Laplace transform Y(3) of the solution of the following initial value problem. y" + 53/ + 6y 2 u(t — 2), y(0) : 0, y'(0) : 0 ~ , n ' ~2‘ 274/33 (ghee g; It {1’ J} 2 ~61 5 “MN-nun... WWMW ? K S : U ’3 w «:2 " 5 T I l ‘3 3“ " ' Z \ , 3-;- f ' if" ‘ \ g" 3 “saliva; 3 l WWNW.WWM_,./ (b)(2pts) Taking the inverse Laplace transform of Y(s) found in part (a) find the , solution of the given initial value problem. 3 U; E i 53/;42’39137‘Kstsr‘3w53 (1&2): mm“; .....,....._...M. 3 1‘2; Jami... 4‘ ,MW WW 7‘ '- M "w ”4 4 .— g siSs‘iligrSE 3 SH, 5+3 3; “‘7 ‘ F“ [cm-wit 3 g ' "’ g s ”‘2 2:? {’2 J 7 <‘ :- «.5. 5* —‘ 6.(2 pts) Find the inverse Laplace transform of F(5) =ln (1+1) 3 Hint: what is the inverse Laplace transform of 9%? 3’; \ "if: ,3 , f \3 «*2 3 3 3 :33: ..... 3:: a. 33:333. 3 "a“; w ,3 is 3 333133.. 3.3 ,3; . i’ w m -3 “7' 3 :3“: j ”:JM 3 3T3 " ,Q, '— 3 {ILL}!- 3; j i 3 i 3' 2"" ’3 3‘35 3 3:5; ......... WWW~WW c 3’ -3 _ ' "‘ 3 ”3;; 3' r” J“ 3 _. «12:6 5’54, 353;; 3“ 533 1“ 3 ’ t ff" j ’ «Wmmww— ‘3 i“ lawn ............................. n 3 7. (a)(2 pts) Find the Laplace transform Y(s) of the solution of the following integral equation using the Laplace transformation and the Convolution Theorem. 15 y(t) + 2/ sin(t -— v)y(v)dv : 5cost O 3%é4m3zg .35 57/33 {we/€53 3313535 (H? 2335/5 5 3 3-3 72:3 ~33 :3 - 2 g a + 33 5 "" 3 M “M“; (7/32"; 3 S 5 3 3 x j 1 «w s 3 \ ’ 3 23. 3 s. ’33 I 3mm“ 33333333333333333 W .......... WW4 (b)(2pts) Taking the inverse Laplace transform of Y(s) found in part (a), find the solution of the given initial value problem. WWMM“_WAM.WWW” .3,.33W3M--M.,,. 33 W“. 3‘ 73 M "r” 3 M 5333 *‘ f” 5:5 3*”), l 5 333332433 t: 9.(2 pts) Find the general solution of the following equation. 4y”—4y’+y=0 3i1 ‘ a? I z" ’1, ,J‘, .5“?va {.A. zdbflq QWyLzMMJj‘L e a (1,. fl ' M (“/31 +~ g! ‘ {j if» i r“ x 2 ,W k} 19% Li $4557!" 5-» z“ 3: M uw—xM”~"“ “f." 2m; a» a, Wt: .4. a: lg E *- i f v 5 Lz‘f‘ Wkfi i r _ WW WWW.” 10.(2 pts) Using Method of Undetermined Coefiicients and Superposition Principle, find the form of a particular solution of the following (non—homogeneous) equation. (you do not have to find numerical value of the coefficients) y”+2y’ +5y: (2t+1)e ‘tsin2t+3t a -. " 2 iv” ’ mu- ‘LW 5 ’“” L/l “1%izgsfi + 5114. 7L _- ’UUVKAWLWJU‘E "(a ; AM \UM . gardfiuufim‘l baton/flew we at“ flW AW Wwfiwm‘ . E K. 2r / i f b 5: A! / X -« 3 4 ‘3 a.» ;-7 I ii’ipfir; “- Era/listings: Cm“ 4” l 1 \JV :3 W t I. ‘2 - i ibis? 39% gamut”? l i {Efflti'f E ,LWWWWWWWWWWWW .WWWWWWW W 11,—Bonus (2 pts) If f, g are piecewise continuous on [0,00), and both of exponential order a, is it necessarily true that L'( f g) : £(f)£(g) (ie. that the Laplace transform of the product is equal to the product of Laplace transforms)? Justify your answer. Hint: Consider f0) : W) = 75- 5"be ilw’MxGU‘iz’y‘L 5' S f‘r’k Cf”? ) gar/vi L6 3: (5’; 6‘": :6 4M ‘4: l l 3‘1 E l . l “W; ,2 \ 2W z; < i .r s. V "L 7’ l‘ g 2? “W ‘ {W 5! 4‘5“: :3 W 3V5 is; - g x X {W 3 uf a” “E j S 3 1m l W ,3 { “3 M: i a? 3 Z “i” ””1“" ;; “in... .3“ {a5 jigs/mu (F194 A i o a .3 W 1 S 1. g L}- - ”W; 1* 3 s = s TEST 2 B (08/05/11) NO CALCULATORS ' m” x , ~ MAP 2302 Name; § 01 0/ .f 04/6 Summer B 2011 3rd period Section 4787 Please use the blank space and the other side of the sheet to write your solutions, clearly numbering them. 1.(2 pts) Find the inverse Laplace transform of 1 / - 1’ § F(s)=ln l—— I)?:, Hint: what is the inverse Laplace transform of %? if: “a «1% f/5 fa -\3 ['2‘ «W 3;: " ’3‘ ":Z ”7‘" (x-trv’lib‘i'”xcu3 a» MW -—-- I S as i 54 5‘ ”4! r7; {- 5 .5!!! g x (vb; fr MW 2):,g m; irivgiifgp :f- Harri/:3 twang”? ii“ w '(r-:\ » 36' 5.5 J, ‘WWMUNMWN as"! j E u! « z i l A,» griffiu {“1“ W L i J L 1“” WWW e a f l, 2. (a) (2 pts) Find the Laplace transform Y(s) of the solution of the following integral equation using the Laplace transformation and the Convolution Theorem. y(t) +/0 sin(t — v)y(v)dv : Boost 4%..- :lvéifl 5,; win. C” 11» five/M (5:4ng .5 3 iv" ’79:»?! (b)(2pts) Taking the inverse Laplace transform of Y(S) found in part (a), find the solution of the given initial yaluewprgblem. ,7,” ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ““37“" 1 WWW ........ Swisher...” r , x, H 5 ’ " z, .r 1‘ 555%; g {z ; é - ~3<ww 3. (a)(2 pts) In the method of variation of the parameter a particular solution of the non—homogeneous equation has the form yp(t) : U1(t)y1(t) +v2(t)y2(t), where y1(t) and y2(t) are linearly independent solutions of the corresponding homogeneous equation. Find 3/106) and y2(t) for the following equation, and set up the equations for U’1(t) and 11505). y” + y : sec3t f ’_ .‘ F‘ A \gQM it 5 gm, ade/ngekm Fl) ‘Mg’i ’{w Viki/”fr“ i) you a Lk’ Y (b)(2 pts) Solving the equations in part (a), find the particular solution of the given equation. 1' ‘ 1! gs- . 13’:ch 513 mm £321,4thch x4 , (iii tfivC/zLJ; f3 ““44 4. (a)(2 pts) Find the Laplace transform Y (s ) of the solution of the following initial value problem. (b)(2pts) Taking the inverse Laplace transform of Y(s) found in part (a), find the solution of the given initial value problem. g a”! with U) 1‘35?” “i“uS‘IfS: ”3:? E 5 4.: 3 A- Z 5 M; ”5-: 3 2 EW/ fl 5‘ was; a ,, fax B 5.(2 pts) Find the Laplace transform of the function defined by v 2 HM D M .W {Acts 19, _. ,. t 7 WNWfiW i WW" *« QC" 1—- $7” ’ , {Z i ." i - If“ ”WM 1; ,’;rF\ i wf); ,' L E hum. v I,“ 375 "'L’ "e w WWW 5»- r". 33’ :5: "“ ;' s‘S-é i .5 “L C suicsi S‘Si i) 5“], ”Susi; ? $5 \ 4’36 g \g :5 E 5CD”) I; L. H i l‘} L mM~w~i~m 6.(2 pts) Find first two non—zero terms of the Taylor polynomial approximation (centered at :50 : 0) of the solution of the following initial value problem (write the corresponding Taylor polynomial): y' + 3sin(:1:y) = 0, 31(0) : 2 PM We; gmgwmi {can 5&4 L«-?t,.-i??‘1”“£“¢7‘3 9'3! £25 Mt 6" ' TM I241 18.11% 020(2 _—~ CH +Q[(o : O ) 04‘ “222w Md 2 I' 4 - I ~ A ... ' ‘ : (2+LY2,1H>L{Q+Q_ (Ra‘DQé-H “*(HQMé (9 Q): I WWW”WWawvmk,,,,_ww_.g.._m.w-mm mmmmm W S g 1 +/é+;\aé+!up/g+g Dag ”€149 Azwmce ( (L+L)(/<%/,>WM/(§0 8.(2 pts) Find the general solution of the following equation. 4y”+4y'+y:0 ’V’/W’”L&(L}Lfli “ii/(“UAW 4/17 é’ U22"? «é ,2 :2: 322; (I ‘2 2, i" 23 22 2 2}; =2: {:2 1 " ’2‘) i 9!. {imp/{.24. 221.2231,’# ,1: :15... ,, I; i . WWWmewWM.WW.W.w»mw.mw~w--m22%’37: 2222222 22222222 2 , 2 «. 2 . :22 " ”iff222,2;222,25_2,z2 { v u . LW:--W W W- WW W W - W -WWWWl 9.(2 pts) Using Method of Undetermined Coefficients and Superposition Principle, find the form of a particular solution of the following (non—homogeneous) equation. (you do not have to find numerical value of the coefficients) y" — 43/ + 5y : (2t + 1)62tsint + 515 272’ 22 222222 “-,. . ,. 2 2. t 2 2 2.) - II . :2 22 W ‘ 32’ EW'IfiW' 2-322222Iw222 c2. 524222.223”! M2 «“2222 mat/W 1 ’ “‘"’ rm“— M“"‘"" ,5 % {ALL ”mi?" f’i'. ‘2“ S 337’ 5:? 2"» ‘2 x" ‘2 2- f. . "a - ’ 22.222222: 2.2.! 2222.23 .22 02-345 1‘ 2 if: .?\\“Z + :3 1‘\ g Féf i ’C 3,1”? k ’1}; ’I‘Aaz J g ’1 v and 2f ., § ”V g” j j . 1e ‘ p7 ' i .a )j _, ’7 «23‘ _ g \ K . U - [$2M $2 2”?“ 3 _,,’ 2L —- 2‘ ‘ n . "- g _ bit. 2 .5. g s 10.(2 pts) Using the Laplace transformation, find 11.—Bonus (2 pts) If f , g are piecewise continuous on [0, 00), and both of exponential order a, is it true that £(fg) = £(f)£(g) (i.e. that the Laplace transform of the product is equal to the product of Laplace transforms)? Justify your answer. Hint: Consider f(t) 2 g(t) : t. :1 I“ 2": f: ” fiance ("m L,” (0216“?! I 29.212222; 21.22:? i 582. {231: . .e 2‘ ...
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