engineering math - 1. (7pts.) Using the Laplace transform,...

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Unformatted text preview: 1. (7pts.) Using the Laplace transform, solve 'the problem. y"+y’ —2'y= 33int—cost if 0 < t <- 211' and 3sin2t—cos2t if t > 211'; y(0) = 1, y'(0) = 0 ML) gifat’-};1=(Mkt'wsi)ut+)—(>74«fhwb*ws£)u(+rho) +0411”, mat) “a in) = ( zmamge) u&) + {a M'b-kawglwm) -~(3aMt-)-I)-w4 (1a.. my] be 65—» Lw) U 55%)} = 9*st) —4 glok'aL’Iv) = 4*Y{s)—; 4H yaw = 9Y(s)—-[ ; d‘ a), __s_ g'z_$ fflfi 9 «3.1,; 5Y4+4Y4 JT'ZZET‘Wr-H Lew 4m; 4’:H+é’=Hj6 L , - ’4'?) 5—5 4—; -m: 44 L [5 ._,I7_H ,____. + F...“ j C9 )Y) “Fm, [gyquf-FH 6 W5), @‘fi'IZfl-fU'Q‘Fé +l‘ (4+6 c.>+l)+u——5)[c.w)j £41,: " (mm-rm (V—H) (9+>)(9-'l)[s’+i)[4*+lp = 91"+€)”f_‘+ + m») (5'1) {9'44 J 09+») 0:1) (9‘4! ) (av—Hp) awn») + mixing) 6*”; 6:14) (4—1) (3+!) (mi/naw) (H4) a .— _ I " ’ “I 4.105 PZ-TJW’?! i4“ fifija WWW 6+- + { wrm2~éewWi MW “"ng JY‘ 2.(4pts.) Find the Laplace transform by differentiation. {flaw/H i: f Ff“) DID/r 'Pi‘fi/ 7—bcoélxrt—b ’U 'Hflj = W9) :.E(ew+g%t) I 3 '7" help? + (étwfij Wu): Ti I ' i try—LP tee-w)“ afli)‘ WW / arr 3.(4pts.) Using the Laplace transform, solve the integral equation. y(:)=1— fort-amen me um vofwfifm, ff Ctr—z) 51(3) pi Z = 4:64 SW?) 359) s [F «6* Edi) MW} = YR) = J;———}:Y(s) We) Ham) = {- 4.(5pts.) Solve the initial value problem by means of Laplace transform. y1'+y2’=2sinht, y2'+y3'=et, y3'+y1'=26t+e_t, 311(0):], y2(0)=1, y3(0)=0 l M: hwa = 7:: éYL‘f‘dY‘r '9’ 2% 1. I 5+! 4Y3 +4Yl-"Pt 7‘" (:LLLI .— )’ L \{l +YL 4(4L_” "—— m __ ,__L_ ’ Y¥+Y5 -‘ F‘Q- 44-4) 40+” . _ > .L l I GHQ ’ LY‘ ’ may!) + ?+ éfL’l) + 44+!) >5‘%)+@+ +£970___%_4_(‘g_fl)__ : L S sub!) "4(4+l)(;—-I) 9*! ‘ r ,__z_ viz-TIA?! ,Y..= [17 " (sle—f)+élé+l)) a 'G"+/¥)~W/| * 9'70 “4&1: J. 5 (4L4) ~ e(4+1)(9——l) 4+! ——(¢‘—;) _ .1; 4‘" ewHJCe—I) ’ flfa—f ){4—1) (B :< H 4. w}... K n i + *3 5.(3pts.) Find the length of thecurve. R13)2 [t, cosht] from t=0 to t=1 1k 4+, F”: U, WM] var/a lepmlft : H IFLCfl/Pt'f‘éhyfid’) :: IFL (at? aha-Hy) = —$[e*+e"b)":wt% I»: f! I am At 2 [was]; = 2 6—5" _ gig CeaMI) ,9 fl._______. 6° "— M, _,——»——'—f 6.(4pts.) Experiments Show that in a temperature field, heat flows in the direction of maximum decrease of temperature T. Find this direction in general and at a point P. T= :cQ-yg, P=.(2,1) an‘l‘an-b HwéAeoé‘thaflMWWaflf 7.(4pts.) Calculate El?) - d? for the following data. If F5 is a force, this gives the work done in the C displacement along C. F: [ea e126}, ?~'= [MW] from (0,0,0) to (2,4,4) F‘C'5)= tf‘ «t w; +49“? Mag“) «twat, wast, ale/w "F’Lmu : 6""? + at"? +31% “tr—9%): '5‘ H—e/j‘ t 1—673 Hello: f: F‘Wflff-Ea’t _ 5 Te" Me‘e‘vj‘wvfif f‘flifijL’I—ltf) d-fi ; f: C 99+» a*’+>t at? (if ‘= [6'5 M‘E‘itatj: 5 el'fifl’g J‘V fiCfi/‘JF': lg‘flfiel—r} M 8.(4pts.) Find the center of gravity (5, E) of a mass of density f(m,ly)= 1 in the regibn R. r) “4%? (M) h M .4"; my 4W =fiff": 015L471, : y wwl—itvl" = it“— 0 r‘l Wharf xPergJJ514°“TJEJLLf:W"JHL% L L L ” ,L L 12— 1 iii" :fijo T9770 “(w-PL [WU 1’ hm" fag—'IO E5" } a ("it "51): (ti—“L, 3%) W ...
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engineering math - 1. (7pts.) Using the Laplace transform,...

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