MTH 205-Solutions of Final Exam -Summer 2007

# MTH 205-Solutions of Final Exam -Summer 2007 - College of...

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Unformatted text preview: College of Arts and Sciences Final Exam — summer 2007 Differential Equations (MTH 205) Instructor’s Name: El 1. Sadek El 8. Khoury Date: July 25, 2007 Student Name: Q ;> Student Number: ———______________ Section'Time: This examination has 10 pages (including a blank paperfbr calculations) plus this cover page. Before you start the examination please verify the number of pages. No Questions are allowed during the examination Student signature: 1. {12 points) Use Laplace transform to solve the! following systmn of dif~ ferential equations: (17.1: *— : —' .~— ‘2 (if. 9+ 6“ ) (ﬂy —~'~ = 4' (it 1 + "U such that ;c(0) = U and MO) = U. 0 .15 0 “(Hr/2%; =3><w> +“‘<~s> -15 SXIS) “‘4' Km) :: e "3 X8) +5305) ﬂung!» :o XIS) .ﬁ. K‘s) : -—l}\ +>E1XI5>+ (gm/1‘05, ta \ \(Ls) -: 36 fr! 5": 1 *2; + 1: 5-") X19 1‘36 i '25 :L-- “r f 5"! 5“; j J g” -' é ’1 { § \$a("c\:§[‘£ 7:? +36 523 £- .— .-— 3 )Us‘) 4-8—1)ng :3 =5; 5 «‘45‘H’ X“) q-ﬂ‘: 33» C? 5' '15") 2 2. (1:3 points) Find each Lapiace Twansjbrm. (i) £[tsi115t] L. _ EL 9' 3 ___ g ('15) __ 1a:- cJS 5"+L5 3 :31 I“ ) 7’{/+;:t2_.{’ur4=-\) +3Ui ‘ S J ‘ 3g; 1 be“)? T . o/{fhlg ": E—g‘ ~— S '2; 3e 1 ,5‘ {tz+;t1—\§ + I. 5 :5,— t L 36 6,} ifzg-Z'f Ik’f‘ _——-—j .— .2“: ---- 5-3 #5 #1. f '5’ 5 +3.3!— 96’5 743—“ “'3'— S jag-'3 —-" 7:.- J —S if“) S '5 23—— + T ‘2‘” J 5” ,. 9.. I} 164(54 +5"; 5) ~._ 7. ,_ (in) u—lw *' 7; x -y 3 - I ~ ‘— 25—‘43: __ﬂ ’- J “If 1“ “b’z’ L5H) 5’59 if t‘é“..z£ é +5 (9—H) 3. (15 points) Find each inverse Lupiacc Tmmﬂnm. (ac—1W]: {'i 534—5 5 rib—H} :14“; * fulfil .3 [9‘75/«679nék4 — J’mfclz: k:( 0 ﬁlm—E [-65.9 ‘ "“bfL (ii) 51%] 2., jﬂlfﬁm —! 9,5 = of f -1 25+G’é :/ -1 J #éz" 5; f :M fw m ri‘éjfws'f _éé3£ﬁn 5'15 4. (10 points) Solve the following differential equation. Give the solution in explicit form. .r y’ + (J3 + 1) y z 9"" 111;:- .4: r. __ C ﬁx ...__’vi__3’-_-_ D +Q+15‘)3 " x w x6:— j{l+ik)ék x443”: 1 f4”) : C 1-: 5 :16 ‘éll) I Ddij'efx' Ak— 5. {H} points) Find the general solution of the following differential equa- tion [Use the undetermined coefﬁcients method to determine the particular solution). If!!! + y” = 1-— .. Mg—tw’" =0 ‘9‘!“ (WH\"‘-’ :53 w:c, Qéa‘bkc> ) “.24 lbs/ll :: C‘+<-2.’C +3ewx- p]; \$3.? :? 76.2217 ‘1 } HDF: {761i}, 9C1} 6. (10 Points) Find the general solution for each of the following homoge— neous differential equations. a. .123 y'” + ary’ = 0. b- y” - 8y = 0. 6 m(m-—I)(w-2\ +h=0 m(h1_3m+18+m =0 m (“1-5M +3) =0 2::J9—fi. _£-;El‘ '3. “:0 m = ‘ I 'l- C“ 3/ m1 1' (ﬁcmofiLQ-f CLSIh {gill “4:1 W1+1“.+Lr:o vim-:1) w:,1_‘F- NIH-'4‘ 7. -— Hi 93— alt-F I'm”... -"+ .pL) W ._ _______.._..—::—— .—- 7. (12 points) a. Use Laplace transform to solve the following integral Equation: 1 I »r If“) 2 if?! — it")! + I 0 (ET -— f3’-T) if“ - T) (37' '* H ‘09):J5 E's—st”) r if/(e-ge )dt‘a) '3 - I .l_ YUM-J— “ 5H + 37*5H)V‘5‘ 3'“; -X I XML-,- 2 +5“; " ‘(H IE»; 5 -| (I '- .3— (Q: 1—...— Sb")\( 5"? x 3 2(;t_1\ Sli— 3 K“) .3 2 1:) (S - (5:3)(5‘L)”+l‘ 5"! :qu 5 + c .2. z [5)- I) (A; +6)(5.t-'1) 1- ((3:0(511) 54,3 I-e 5+; +D(S‘l—3){5-L) 5“ =9 ( -..L,o =) arm Era-1 a: 6 :41) 737 “=41; (FMsM-n :5 J; *‘5 “V S=ﬁ=§q=€rhﬁfﬂé 3-6:“! = - 1' ' “HG g) H 3 W—zé =9 358:“! Jr“ =‘1 => h:o 2.1g waft ' am =43. mm - 2;} .. ac b. Find the inverse Lapﬁace Transfoma. WMSBH” 11:, ‘4 Fm} Jiifmj :- “éFM =9 1PM =3" {as u 8. [12 points) A mass weighing ‘2“ pounds stretches n .5- ;m‘ing foot. The mass is initiaity released from a point ft below the eqniiib-r‘inin position with. or, downward Unionity oféi ft/o. 5:. Find the eqnotion of motion. 13. Deteﬁnine the period. freqnency, and omptitnde of the motion. 0. F ind the iricutimnm displacement of the 3 ping from the egnitibiinm po- .sition. d. At what. time does the mass pass thran ing downward for the second time? oh the equilibrium position head- 9. {1i} ].}oi11ts) Solve- thc :i-iﬂe'r-ceutmi equut‘éu-nh Grim: the soﬁution m explicit .. z“ 3 ‘ 2:17'yy'=4;53+3y2 # If)” '(‘t r 3 form. =5 u’uéu = (“NH-11353" 77" MNAW r: xILLH'u“)At a. an “I :- ﬁAL +L u‘--H .Q Woo-H) :: thdfc (=3; u‘+"\ '5 A1 a) u“ :1 Pm " s’) u r '1 W =17 3:; = T?" \J P” ﬁr :: '51" 3‘ J k1 rt! 395 ...
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