Diaforikes2004 - TMI[MA MEKTPDADFQN MHXANIKQN KAI MHXANIKQN...

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Unformatted text preview: TMI-[MA MEKTPDADFQN MHXANIKQN KAI MHXANIKQN YHOAOFIZTQN TEAIKH EEETAEH 3”" EEAMHNGY ETIE fiIAIIlOPIKEE EEIEflEEIE 'iIlEBPUYAPIUE Zflfl-fil ZHTI-HHA DPflTfl: (a) Na. unnfimxflei Em av y] 1cm y: Exam Em ané impair: mum‘lg In (rm fiuiarnuu I , 115115 55%.! 111:0an v»: 5:11:01an Eva Bapeluhfiag £113qu 11353011.: mg Slurpnpucfig Efiiamm'lg y” + p{r)y’ + q(f)y = U UTE! fitcifirnpcc tau-cf}, 51min; 5:211! 01, mwalamég pm 1cm quit) pnfiafifiuwm mm In. (3 pmfifiag} (b) Esmpnfiua 'n'lv efiinmc'n'l Euler xiy" + Easy“ + fly: 0. Na Bpsfiofiv m mvflfiwag mu npmat w: :11an m mahfitég :1 m1 ,3 mats £513; 01 luring 111g Eéiqmrmg vu Eivm (@quth mamg x —> n . ' (4 pflvfitfiag} 'ZHTHMA fiEYTEPO: (a) Not b.1851“. tn npfifllnpu'upxmfiv 111.1611: fl 3: , 0 1‘; r «a: 2 y +2.? = gm , git} = G , r E 2 M0) = 0 , y’w} =1 [6 povflfiag) (h) Emu: Em. 1] f £3551 m; {1va 111 mmmvucfi Gama Fflurier ' f{I]- ‘_ 2—"+ 2a,, cos-E. Nu umfislxflai m1. % fl f{x)]1dx — 2§+Zaj "-1 _ {2 puvflfiag) (a) Na. xupuxmpmfisi m uvfinpufim mumin 111;, (J: #1)}? + fixzy’ + 3 y = 0 . 21111.: napimmdn mm and: shut Kmnwxé mfdmuln await: w: Bpaflai T] fiaimtptu aim-mm] Km 01 exflétsg fimunptpiug 1cm w viva: Empnm] Tm Tn afiwclmn 111:; Gaping um: lfinsmv. {5 povifiag) ZHTI-II'PIA TPITO: (a) Na BpEitE “cm! afiifimm m1 w: npnnfimpiaata ID Eifiog mg mun-fling 1:01.} (Ivfiflfit mu mmyéwsm: (2x+y)d§r+(2y+ x)dy = 0 Km napwial :1me TD music: {—1.2} 1:01: Eméfiou (bay. (3- nflvflfisq} (b) Nu npmfimpifiam 17:1 mimpa mpfiifl. mm Hg lflamg moppnnim; mg fimcpnpmfig 55:51:}:qu 23—)- = a" ‘1 5:61“ 2111 fluvéxam w. Bpsirs Tn Tammi 113511 “1:11;; afiifimfing unn‘lg 1cm TD épm may lfinamv Tm: : —3- +0: . T1. mupaita vu Unmapcivstm Tm Tn him-l yfi) = 0, VI E [R m»; :rtpm; rm: aumftflstfl mg; {7" Hflvflfifiq) ZHTHMA TETAPTD: (a)NI1 fipafiai 11 yawn"; film] 1119, Emmapucfig Eéifimfing . 1 . )4 +Ey—3Slfl2f=fl,f}fl ,y=y{f). (4 Hfivfifiggj {I1} Amflmwmcfl Kill pfivu LIE In péflnfin 1cm: fitnnmhv w, Bpeita :11 113cm mu npnfllfipumg Emmy npcbv 1011 w: fipflfi‘filflpififitfi my mm 111:; mpflmunng I12 + 3:; Tu: TI] 3.13m} [1111f]. x; = ~x1 —sim‘ , x; = x1+cnsn II = 1,0),11 = x1 (I) xl(fl) =1 , 352(1)): 0 {6 unvflfiaq) METAEXHMATIEMDI LAPLACE I ” fl 3 L {m _ mm: — {1)} = e'fl {my} LIEEDSHI): 1 ., 3—H 3 _-3.5 +cr“ 3'+r::r TMHMA HAEKTPOAOFQN NHJXANIKQN KAI Nfl-IXANIKQN YHOAOFIETQN EIIANAAHHTIKH EEETAZH 3°” EEAMHNOY ZTIZ AIAQOPIKEZ‘. EEIEQEEIE ZEHTEMBPIOE 2004 ZHTHMA IIPQTO: (a) Na Bpsesi ’CO Gaushcbfieg Gévoko Mceow mg avriormxng ouoysvofig mg 5.8. 1‘2 y” — 2y = 31‘2 —1 , t > 0 , ow T] y1 (1‘) = t2 given pm X13611 mg opoyevofig, Km va 5068i 11 ysvucr’] Mm] mg m opoygvofig. , (5 uovdSsg) (b) Av y = (p0) given pm Mon mg y” + p(t)y' + q(t) y = g(t) , g(t) ¢ 0 , VOL sgetac'cei ow n y = C(p(t) , C = chspo’c , C 921, aivm Mm] mg sgicmq'ng ommg. (2 uovdfieg) (c) Av 11 p(t) > 0 given 51amopicmm VOL Dnokoytcstai n opiCoucu Wronsky mg 5.8. (p(r)y')' + 42(0750- (3 uovdéag) ZHTHMA AEYTEPO: (a) Na Xveai p8 xpficm TOD pamoxnuomoum’) Laplace Io npéfiknpa , n , , sint , 0 s t < 1: apxmwv “new y +y =g(t),y(0):0,y(0)=0 ,g(t)= 0 t)“ (5 uovdSag) (b) Na npoofitoplcyrai éva 1(de (ppéwua mg aKrivag cfiywhcng TCOV Mascov GS cetpd 76pm and w (musia x0 = 0 Km x0 = 2 7w mv sgiccocm (1+ x3)y" + 4xy’ + y = 0. (2 uovdfieg) x , 0 S x <1 } (c) Na BpsOei TO nunovucé owdnwyua Fourier mg f (x) = {1 1< 2 . H01; cmflivst T] , _ x < 6231de “ (3 uovdéag) ZHTHMA TPITO: (a) Na X60818 TO npéBKnua apxuccbv nucbv ,_ y3 ~3x2y+2y y x3 ~3xy2 —2x ay=y(X),y(1)=2 (5 uovdSag) (b) Na npooStopiosrs m Guvétpmon y(x) 7w mv 07mm LUXISSL r3 + (r mm = [xy'<x>dx ,ym) = 0 (Nu m] xpnmuonomeai usmcxnpomouég Laplace) (5 uovdSeg) ZHTHMA TETAPTO: (a) Na BpaGofiv (Slag 01 kfioelg mg Stacpopucr'lg sgiococmg y(y’)2 + (x-y)y’ - x = 0 1cm VOL npoonpwrofiv axeivsg 7:01) Stépxovmt and to (music (1, O) . (3 povdfiag) (b)A7IOK7\,816'ClK('l Km pévo p8 m ”(6.6050 103v Lfitonpdw vet Bpaits m X1361] TOD npoBMuatog apxmcbv nwbv 2 xf+2xl+4x2=1+4t,x§+xl—x2= I, xl=xl(t),x2=x2(t) Nib) x,(0) = 2 , x2(0) = 3 (7 uovdSsg) METAZXHMATIEMOI LAPLACE I]! (ll 1 S . a n , .. 5"” , LU” )2 5—H) Meow?) 2 S: +517: , L(_smat) = S: +a: , L[H(2‘ —a)f (I —a)] = e“"“L[f(I')] KAAH EIIITYXIA L0") : ...
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