SOLUSI ANALITIK DAN SOLUSI NUMERIK PERSAMAAN CAUCHY-EULER SKRIPSI

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1 SOLUSI ANALITI DAN SOLUSI NUMERI PERSAMAAN CAUCHY-EULER SRIPSI Oleh: INAYATUL HASANAH NIM. 5 JURUSAN MATEMATIA FAULTAS SAINS DAN TENOLOGI UNIVERSITAS ISLAM NEGERI MALANG MALANG 9

2 SOLUSI ANALITI DAN SOLUSI NUMERI PERSAMAAN CAUCHY-EULER SRIPSI Djk epd : Uversts Islm Neger Mlg Utk Memeh Slh St Persrt Dlm Memperoleh Gelr Srj Ss (S.S) Oleh : INAYATUL HASANAH NIM. 5 JURUSAN MATEMATIA FAULTAS SAINS DAN TENOLOGI UNIVERSITAS ISLAM NEGERI MALANG MALANG 9

3 SOLUSI ANALITI DAN SOLUSI NUMERI PERSAMAAN CAUCHY-EULER SRIPSI Oleh: INAYATUL HASANAH NIM. 5 Telh dperks d dsetj tk dj Tggl: 4 Jr 9 Dose Pembmbg Mtemtk Dose Pembmbg egm Drs. H. Trmd M. S NIP Mrl Abd M.Ag NIP.5 64 Megeth et Jrs Mtemtk Sr Hr M. S NIP. 5 8

4 LEMBAR PENGESAHAN SOLUSI ANALITI DAN SOLUSI NUMERI PERSAMAAN CAUCHY-EULER SRIPSI Oleh: INAYATUL HASANAH NIM. 5 Telh Dperthk d dep Dew Pegj Skrps d Dtk Dterm Sebg Slh St Persrt Utk Memperoleh Gelr Srj Ss (S.S) Tggl 9 Jr 9 Ss Dew Pegj Td Tg. Pegj Utm : Usm Pgl M. S ( ) NIP et : Sr Hr M. S ( ) NIP Sekretrs : Drs. H. Trmd M. S ( ) NIP Aggot : Mrl Abd M. Ag ( ) NIP Megeth d Megeshk et Jrs Mtemtk Sr Hr M. S NIP. 5 8

5 SURAT PERNYATAAN Yg bertd tg d bwh : Nm : Itl Hsh NIM : 5 Fklts : Ss d Tekolog Jdl Skrps : Sols Altk d Sols Nmerk Persm Cch-Eler Metk bhw skrps tersebt dlh kr s sedr d bk kr org l bk sebg mp keselrh kecl dlm betk ktp g telh dsebtk smber. Seljt pbl dkemd hr d klm dr phk l bk mejd tggg jwb Dose Pembmbg d/t Pegelol Fklts Ss d Tekolog Uversts Islm Neger (UIN) Mlg tetp mejd tggg jwb s sedr Demk srt pert s bt deg seber-ber d pbl pert tdk ber s bersed medpt sks kdems. Mlg 4 Jr 9 Yg metk Itl Hsh

6 ôô çç ÎÎÎ ôô ãã øø $$ tt ÎÎ ôô çç ÎÎÎ ôô ãã øø $$ tt ÎÎ ss MOTTO #Z ZZ Z ô ô ç ç Î ô ô ã ãèø ø9$ $# ìt tβ βî Î) # ô ô ç ç Î ô ô ã ãèø ø9$ $# ìt tβ βî Î*s sù re Sesggh sesdh keslt t d kemdh Sesggh sesdh keslt t d kemdh. (QS: Al Isroh 94:5-6)

7 Utk Bpk M. Ftch Ib Mzro h Adkk M. Fjr Whd d A Rswd Upe Urfh Me g sell mesport pels dl segl hl d Seseorg g telh memlk seth ht pels

8 ATA PENGANTAR Alhmdlllhrrobbl lm segl pj skr ke hdrt Allh SWT ts lmph rhmt tfq d hdh-n hgg pels mmp meelesk pels skrps g berjdl Sols Altk d Sols Nmerk Persm Cch-Eler" deg bk. Sholwt sert slm semog sets tercrhk kepd jjg Nb besr Mhmmd SAW sebg swt hsh dlm merh keskses d d d khrt. Pels medr bhw bk phk g telh berprtsps d membt dlm meelesk pels skrps. Oleh kre t rg do d cp term ksh g sebesr-besr pels smpk tertm kepd:. Prof. Dr. H. Imm Sprogo selk Rektor Uversts Islm Neger (UIN) Mlg.. Prof. Dr. Stm Bmbg Smtro SU. D.Sc selk Dek Fklts Ss d Tekolog Uversts Islm Neger (UIN) Mlg.. Sr Hr M.S. selk et Jrs Mtemtk Fklts Ss d Tekolog Uversts Islm Neger (UIN) Mlg. 4. Drs. H. Trmd M.S selk dose pembmbg g telh melgk wkt tk memberk pegrh selm pels skrps. 5. Mrl Abd M.Ag selk dose pembmbg kegm g telh memberk sr d bt selm pels skrps.

9 6. Ar smstt S. S g sell memberk msk d motvs kepd pels. 7. Selrh Dose Fklts Ss d Tekolog Uversts Islm Neger (UIN) Mlg g telh memberk lm pegeth kepd pels selm d bgk klh sert selrh krw d stf UIN Mlg. 8. Ahd d Ibd terct g td lelh memberk do d ksh sg sert keperc. 9. Tem-tem mtemtk gkt besert sem phk g telh bk membt dlm peeles skrps. Akhr semog skrps dpt bermft bg kt sem. Ame. Mlg Jr 9 Pels

10 DAFTAR ISI HALAMAN JUDUL HALAMAN PERSETUJUAN HALAMAN PENGESAHAN SURAT PERNYATAAN MOTTO HALAMAN PERSEMBAHAN ATA PENGANTAR... DAFTAR ISI... DAFTAR TABEL... v DAFTAR LAMPIRAN... v ABSTRA... v BAB I PENDAHULUAN..... Ltr Belkg..... Rms Mslh Bts Mslh Tj Peelt Mft Pels Metode Peelt Sstemtk Pembhs... 7 BAB II AJIAN TEORI Persm Dferesl Persm Dferesl Ler.... Opertor Dferesl....4 Mtrks Determ d Atr Crmer....5 ombs Ler ebebsler d Nl Wrosk Mslh Nl Awl Metode Vrs Prmeter... 8

11 .8 Metode Peeles Persm Dferesl Ler Homoge... A. Peeles PDL Homoge deg oefse Vrbel dlm Betk Persm Cch-Eler....9 Metode Nmerk Peeles PDB secr Nmerk Metode Rge-tt Glt j egm BAB III PEMBAHASAN Sols Altk Persm Dferesl Ler Cch-Eler Sols Nmerk Persm Dferesl Ler Cch-Eler Alss Glt Persm Dferesl Ler Cch-Eler Tj Agm Terhdp Hsl Pembhs BAB IV PENUTUP esmpl Sr... 9 DAFTAR PUSTAA LAMPIRAN

12 DAFTAR TABEL No Jdl Hlm.. Besr Glt Atr Metode Rge-tt deg Metode Vrs Prmeter PDL Cch-Eler orde Besr Glt Atr Metode Rge-tt deg Metode Vrs Prmeter PDL Cch-Eler orde

13 DAFTAR LAMPIRAN No Jdl. Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge- tt Orde 4 Meggk Softwre Mtlb. Otpt Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb. Grfk Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb 4. Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge- tt Orde 4 Meggk Softwre Mtlb 5. Otpt Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb 6. Grfk Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

14 ABSTRA Hsh Itl. 9. Sols Altk d Sols Nmerk Persm Cch-Eler. Skrps Jrs Mtemtk Fklts Ss d Tekolog Uversts Islm Neger Mlg. Pembmbg: (I) Drs.H. Trmd M. S; (II) Mrl Abd M. Ag t c: Persm Cch-Eler Metode Vrs Prmeter Metode Rge-tt Persm dferesl merpk model mtemtk g ckp petg. Persm dferesl dlh persm g memt tr st (t beberp) fgs g tk dketh. Jes persm dferesl sgt bk sekl bk dlht dr betk order koefse mp keler sehgg bk jg cr meelesk. Slh st betk persm dferesl ler deg koefse vrbel dlh persm Cch-Eler g dpt dselesk secr merk mp ltk. Skrps bertj tk mejelsk sols ltk d sols merk persm dferesl ler tk homoge deg koefse vrbel khss persm Cch-Eler. Pels skrps meggk metode kj ltertr t kepstk. Secr ltk pels meggk metode vrs prmeter sedgk secr merk pels meggk metode Rge-tt orde 4 deg bt softwre Mtlb. Dr hsl lss d pembhs mejkk bhw sols secr ltk d hsl ters secr merk tk persm dferesl ler Cch-Eler sm-sm medptk l tetp besr l tr ked sols t terdpt selsh g dsebt glt. Jd sols ltk sebg pegotrol glt tk sols merk. Besr glt pd peeles persm dferesl ler Cch-Eler k tetp kek setp lgkh tdk terll besr sehgg tdk terll mempegrh pertmbh keslh. Sehgg keslh pd persm dferesl ler Cch-Eler stbl. Hl berrt bhw metode Rge-t orde 4 merpk metode g l medekt sols eksk.

15 BAB I PENDAHULUAN. Ltr Belkg Berkembg berbg lm pegeth telh bk memberk smbg pemkr tk dpt meelesk berbg permslh g mcl. Dlm perkembg d kemj lm pegeth d tekolog lm mtemtk terp memberk smbg g besr dlm pemech mslh. emj tekolog g sgt meggmk dews tdk mgk dpt terjd tp bt mtemtk. Mtemtk memp per g tm t mtemtk memberk cr berfkr g jels logs tept d kosste sebg sr pegembg pegeth. Mtemtk t pd dsr berkt deg pekerj meghtg sehgg tdk slh jk kemd d g meebt mtemtk dlh lm htg t lm hsb (Abdsskr 7:8). Bk t-t Al Qr g bers tetg perhtg t mtemtk. Sebgm dlm srt Al- hf t 5 d srt Al-Akbt t 4: $Yèó Î@ (#ρßš#šø $#ρš ÏΖÅ 7πs ($ÏΒ] roóοîγï ôγ. Îû(#θèWÎ6s9ρ Art: D merek tggl dlm g merek tg rts th d dtmbh sembl th (lg) (Q.S Al-hf: 5) ãνèδ s{rsù $YΒ%tæ š Å Ηs~ ωî) >πζ #ø9r& öνîγ Ïù ]Î7sù ϵÏΒöθs% 4 <Î) % θçρ $Ζù ö r& ô s)s9ρ tβθßϑî sß öνèδρüχ$sùθ Ü9$#

16 Art: D Sesggh km Telh megts Nh kepd km Mk tggl d tr merek serb th krg lm plh th. Mk merek dtmp bjr besr d merek dlh org-org g zlm (Q.S. Al-Akbt: 4 ) Dr ked t d ts dpt dketh bhw Al-Qr telh berbcr tetg mtemtk. osep mtemtk g dsebtk dr t tersebt dlh kosep blg opers pejmlh opers pegrg. Adp mk g tersrt dblk t tersebt dlh setp mslm perl memhm tetg blg d opers blg. Bgm mgk seorg mslm dpt megeth bhw Ashbl khf tertdr d dlm g selm 9 th jk tdk dpt meghtg 9 jg bgm mgk seorg mslm dpt megeth bhw Nb Nh tggl bersm km selm 95 th jk tdk dpt meghtg -5 (Abdsskr 6: 6-6). Slh st cbg dr mtemtk g termsk topk petg tk dbhs dlh persm dferesl. Topk dgk tk memechk mslh-mslh g dhdp dlm bdg ss d tekk. Persm dferesl membt memechk mslh-mslh dlm bdg tersebt. Dlm ss d tekk serg dtemk mslh-mslh g peeles tdk dpt dts deg h meggk rms t kosep g d. Terdpt bk feome-feome g melhrk model mtemtk m model mtemtk megdg lj perbh. Dlm sts sepert dbthk peeles t perhtg mtemtk secr khss. Perhtg-perhtg mtemts tk memechk mslh-mslh tersebt memerlk persm dferesl (smh 989:)

17 Persm dferesl dlh sebh persm g megdg dervtf/dferesl dr st t lebh vrbel terkt terhdp st t lebh vrbel bebs (Bdr :). Persm dferesl g terbetk dr permslh g d tersebt jg bermcm-mcm. Ad d mcm persm dferesl t persm dferesl bs d persm dferesl prsl. Berdsrk betk terdpt persm dferesl homoge d persm dferesl tk homoge. D smpg t berdsrk orde (tgkt)- terdpt persm dferesl orde st persm dferesl orde d persm dferesl orde tg smp deg persm dferesl orde- (orde tgg). Sedgk berdsrk koefse terdpt persm dferesl deg koefse kostt d persm dferesl deg koefse vrbel (pebh). Sert berdsrk keler terdpt persm dferesl ler d persm dferesl tdk ler. Oleh kre bk jes persm dferesl mk bk pl cr mecr peeles. Peeles persm dferesl dlh st fgs g memeh persm dferesl d jg memeh kods wl g dberk pd persm tersebt (Azz 6:5). D dlm peeles persm dferesl secr lts bs dcr peeles mm g megdg kostt sebrg d kemd megevls kostt tersebt sedemk sehgg hsl ses deg kods wl. At deg kt l sols ltk merpk sols kot sehgg sols dr l vrbel bebs dpt dtemk sgt krt d tept (Lm 994:). Metode peeles persm dferesl secr lts terbts pd persm-persm deg

18 betk tertet d bs h tk meeles persm ler deg koefse kost. Metode peeles merk tdk d bts mege betk persm dferesl. Peeles g dperoleh berp ters merk dr fgs tk berbg vrbel bebs. Peeles st persm dferesl dlkk pd ttk-ttk g dtetk secr berrt. Utk medptk hsl g lebh telt mk jrk (tervl) tr ttk-ttk g berrt tersebt dbt semk kecl. At deg kt l sols ltk dlh seles g memeh persm seml secr eksk sedgk merk dlh seles g berp hmpr (Ssl 99:). Mert pdg slm setp mslh d beberp peeles g dpt dmbl jl kelr t sols pemech dr st mslh. etk st mslh t slt tk dselesk deg st cr mk hl tersebt pst d cr t peeles g l. Sebgm dlm frm-n pd l-qr srt Al-Isroh t 5-6: #Z ô ç Î ô ãèø9$# ìtβ βî) # ô ç Î ô ãèø9$# ìtβ βî*sù Art: re Sesggh sesdh keslt t d kemdh (5) Sesggh sesdh keslt t d kemdh (6). (Al-Isroh: 5-6) Berdsrk ltr belkg dts mk pels meggkt permslh tetg Sols Altk d Sols Nmerk Persm Cch-Eler

19 . Rms Mslh Berdsrk ltr belkg dts mk rms mslh g k dgk t:. Bgm meetk sols ltk persm Cch-Eler?. Bgm meetk sols merk persm Cch-Eler?. Bgm perbdg hsl peeles dr sols ltk d sols merk persm Cch-Eler?. Bts Mslh Adp bts mslh dlh sebg berkt:. Persm dferesl g dpk berjes persm dferesl tk homoge orde-.. Sols ltk deg meggk metode vrs prmeter.. Sols merk deg meggk metode Rge-tt orde-4..4 Tj Peelt Berdsrk rms mslh dts mk tj pels skrps dlh. Medeskrpsk sols ltk persm Cch-Eler.. Medeskrpsk sols merk persm Cch-Eler.. Medeskrpsk perbdg hsl peeles dr sols ltk d sols merk persm Cch-Eler.

20 .5 Mft Peelt Adp mft dr pels skrps dlh:. Bg pels sebg tmbh forms d wws mege bgm meetk sols ltk d sols merk persm Cch-Eler.. Bg pemerht mtemtk sebg msk d smbg pemkr tk memechk permslh dlm meetk peeles persm dferesl ler tk homoge deg koefse vrbel meggk metode g lebh efse.. Bg lembg UIN Mlg tk bh kepstk g djdk sr pegembg wws kelm khss d bdg mtemtk..6 Metode Peelt Metode merpk cr tm g k dtemph tk meemk jwb dr st permslh. Metode peelt g dgk dlm skrps dlh std ltertr. Std ltertr dlh peelt g dlkk deg bt bermcm-mcm mterl g terdpt d rg perpstk sepert bk mjlh dokme ctt ksh-ksh sejrh d sebg (Mrdls : 8). Adp lgkh-lgkh mm dlm pels skrps dlh megmplk dt-dt d forms deg cr membc d memhm ltertr-ltertr g berkt deg kj persm dferesl t

21 megkj bk-bk dtr dlh Persm Dferesl Bs deg Peerp Moder Persm Dferesl d Mtemtk Pemodel sert beberp rtkel g terkt deg pembhs g d. emd megtk kj-kj tersebt deg pokok bhs pels skps..7 Sstemtk Pembhs Agr pels skrps lebh terrh mdh dtelh d dphm mk dgk sstemtk pembhs g terdr dr empt bb msgmsg bb dbg ke dlm beberp sbbb deg rms sebg berkt: BAB I PENDAHULUAN Pedhl melpt: ltr belkg permslh rms mslh bts mslh tj peelt mft peelt metode peelt d sstemtk pembhs. BAB II TINJAUAN PUSTAA Bg terdr ts kosep-kosep (teor-teor) g medkg bg pembhs. osep-kosep tersebt tr l membhs tetg Persm Dferesl Persm Dferesl Ler Opertor Dferesl Persm Dferesl Ler Homoge deg oefse ost Mtrks Determ d Atr Crmer ombs Ler ebebsler D Nl Wrosk MNA Metode Vrs Prmeter Peeles PDB secr Nmerk metode Rge-tt d Glt sert j egm.

22 BAB III PEMBAHASAN Pembhs bers tetg sols ltk sols merk lss glt d keterkt tr hsl peelt deg kj kegm. BAB IV PENUTUP Pd bb k dbhs tetg kesmpl d sr.

23 BAB II AJIAN TEORI Pd bb k dberk beberp defs d teorem g berhbg deg pejels pd bb seljt.. Persm Dferesl Defs. Persm dferesl dlh persm g megkt tr dr st t lebh vrbel tk bebs terhdp st t lebh vrbel bebs. (Ross 984:) Berdsrk jmlh vrbel bebs persm dferesl dkelompokk mejd persm dferesl bs (PDB) t Ordr Dfferetl Eqto (ODE) d persm dferesl prsl (PDP) t Prtl Dfferetl Eqto (PDE). Defs. Persm dferesl prsl (PDP) dlh persm dferesl g megkt tr prsl dr st t lebh vrbel tk bebs terhdp st t lebh vrbel bebs. (Ross 984: 4) Cotoh: v v ) v s t ) z

24 Vrbel bebs pd cotoh dlh s d t sedgk vrbel tk bebs dlh v. Seljt pd cotoh vrbel d z dlh vrbel bebs sedgk vrbel dlh vrbel tk bebs. Defs. Persm dferesl bs (PDB) dlh persm dferesl g megkt tr bs dr st t lebh vrbel tk bebs terhdp st vrbel bebs. (Ross 984: 4) Cotoh: d d ) d d 4 d d ) 5 s t 4 dt dt Pd cotoh vrbel dlh vrbel bebs d dlh vrbel tk bebs. Sedgk pd cotoh vrbel bebs dlh t d vrbel tk bebs dlh. Defs.4 Orde st persm dferesl dlh orde (tgkt) tertgg dr tr g terdpt pd persm tersebt g tgkt plg tgg. (Pmtjk d Stos 99: ) Pd beberp cotoh dts cotoh dlh persm dferesl g berorde st sedgk cotoh dlh persm dferesl berorde d.

25 . Persm Dferesl Ler Sepert g telh djelsk pd sbbb. persm dferesl ler dpt dbg mejd persm dferesl ler homoge d persm dferesl ler tk homoge. Agr lebh mdh berkt dberk defs dr persm dferesl ler persm dferesl ler homoge d persm dferesl ler tk homoge. Defs.5 Persm dferesl ler berorde dlh persm dfreesl g memlk betk mm: ( ) ( ) ( ) ( ) ( ) b( ) (.) deg d b fgs-fgs kot pd tervl I g h bergtg pd sj d ( ) dperoleh:. (Ross 984:5) Apbl msg-msg rs pd (.) dbg deg ( ) mk ( ) ( ) f ( ) f ( ) g( ) f (.) deg f f d b g betk (.) dsebt sebg betk bk persm dferesl ler. Cotoh: d d ) 5 6 d d ) 4 d 4 d d d d d e

26 Defs.6 Persm dferesl ler homoge dlh persm g memeh (.) deg b ( ). Betk mm persm dferesl ler homoge t: ( ) ( ) ( ) ( ) ( ) (.) d persm g memeh (.) deg ( ) b dsebt persm dferesl ler tk homoge g betk mm dlh sepert pd (.) deg ( ) b. Persm (.) merpk persm dferesl ler homoge g berses deg persm dferesl ler tk homoge (.). Hl k dgk dlm meelesk persm dferesl ler orde- pd bb III. Defs.7 Jk sem koefse ) ( ) ( ) dlh tetp mk persm ( (.) dsebt persm dferesl ler deg koefse kostt. D jk sem koefse ) ( ) ( ) dlh berp fgs mk persm ( (.) dsebt persm dferesl ler deg koefse vrbel (pebh). (Fzo Lds988:58). Opertor Dferesl Dlm pembhs persm dferesl tdk leps dr opertor dferesl. Berkt djelsk mege opertor dferesl. Perhtk rs kr dr (.) t ( ) ( ) ( ) ( ) ( ) (.4)

27 Jk dlh fgs g memlk tr ke- mk (.4) dpt dtk deg: L [ ] ( ) ( ) ( ) ( ) ( ) (.5) L pd (.5) dlh fgs g br g ddefsk pd hmp fgs g memlk tr ke-. D s L dsebt sebg opertor. D kre L melbtk dferesl mk L dsebt opertor dferesl (Sswto 997: ). Jk ddefsk D ( ) Mk (.5) dpt dtls sebg: L ( ) [ ] ( ) D ( ) D ( ) D ( ) D opertor dlh: (.6) ( ) ( ) D ( ) D ( ) D ( ) L (.7) Jk dlh kostt mk opertor dtls deg: ( ) ( D) D D P D (.8) Mert (Goode 99:6) opertor st persm dferesl ler membetk st trsforms ler..4 Mtrks Determ d Atr Crmer Metode peeles g dgkpk pd bb seljt berhbg deg determ mtrks. Dbwh dberk pegert dr determ st mtrks.

28 Defs.8 Jk dketh mtrks A t A mtrks A dlh det ( A) Cotoh: Htglh determ dr mk determ dr. (Ato 997:6) A 4 Deg meggk defs dts mk k memberk det Defs.9 ( A ) ( )( ) ( )( 4) Jk dketh mtrks perseg A mk determ A dpt dhtg deg meggk mor-mor M j dr A dm mor M j dr A merpk mtrks ( ) ( ) kolom ke-j dr A. (Ato 997:79) Perhtg determ dlh: g dperoleh deg mecoret brs ke- d det j j ( A) M M ( ) j det ( Mj ) ( ) j det ( Mj ) L L j (.9) Cotoh: Mslk A Htglh det (A).

29 Peeles: det( A ) () (-4) - ()(-) - Utk megeth d d tdk vers st mtrks perseg dpt jg dgk determ mtrks. Berkt dlh teorem g merpk pegj terhdp keberd vers st mtrks perseg. Teorem. Bkt: Mtrks perseg A memlk vers jk h jk det ( ) Jk A dpt dblk mk ( ) det ( A) det ( ) det A det ( A ) I. Jd det ( A ) A. I AA sehgg. Seblk ggplh bhw. t k perlhtk bhw A ekvle brs pd I d deg demk kt dpt memplk bhw A dpt dblk. Mslk R dlh betk eselo brs teredks dr A. re R dpt dperoleh dr A deg meggk rt berhgg dr opers brs elemeter mk kt dpt mecr mtrks-mtrks elemeter E E Ek sehgg E E k E A R t k A E E E R. Jd det ( A) det ( E ) det ( E ) det ( Ek ) det ( R)

30 re kt megggp bhw det ( A ) dperoleh bhw det ( R ) ol sehgg deg demk Cotoh: re brs pertm d brs ketg dr mk dr persm. Mk R tdk memp st brs blg R I. Jd teorem terbkt. (Ato 997:74-75) A 4 6 sebdg mk det ( A ). Jd A tdk dpt dblk. Utk mecr seles st sstem ler dperlk tr perhtg tertet. Slh st dlh deg meggk tr Crmer g memftk determ mtrks. Atr Crmer dgk tk meelesk sstem ler g memt persm d pebh. Atr Crmer dtgk dlm teorem berkt. Teorem. Msl dketh sstem persm ler dlm vrbel t: M b b b (.) deg j d b kostt vrbel j. Jk mtrks koefse dr sstem (.) t A M L L M memlk determ tk ol mk sstem (.) memlk seles tggl t:

31 A A ) det( ) det( d m A dlh mtrks g kt dptk deg meggtk etr-etr dlm kolom ke dr A deg etr-etr dlm mtrks b b b B M Bkt: Jk ) det( A mk A dpt dblk. D XA - B dlh pemech tggl dr AXB. Sehgg dperoleh: b b b C C C C C C C C C A B A dj A B A X M M M M ) det( ) ( ) det( Deg meglk mtrks-mtrks k memberk C b C b b C C b C b b C C b C b b C A X M ) det( Etr dlm brs ke j dr X deg demk dlh ) det( A C b C b b C j j j j

32 Sekrg dmslk j j j j j j j b b b A M M M M M M re A j berbed dr A h dlm kolom ke j mk kofktor dr etr-etr b b b dlm A j dlh sm sepert kofktor dr etr-etr g berses dlm kolom ke j dr. Perls kofktor det (A j ) sepjg kolom ke j deg demk dlh j j j j C b C b b C A ) det( Deg mesbsttsk hsl ke dlm ) det( A C b C b b C j j j j mk k memberk ) det( ) det( A A j j terbkt. (Ato 997:8-84) Cotoh: Gk tr Crmer tk memechk Peeles: 6 4 A A

33 A A Mk det( A ) 4 det( A) 44 det( A 7 ) det( A) 44 det( A ) 5 det( A) ombs Ler ebebsler D Nl Wrosk Dlm peeles persm dferesl ler tk homoge deg meggk metode vrs prmeter dperlk pegeth tetg kebebsler d l Wrosk. Metode vrs prmeter k dbhs pd bb seljt. Berkt dberk defs d teorem tetg kombs ler kebebsler d l Wrosk. Defs. Jk dketh hmp vektor f f f c c c dlh d kostt mk betk c f c f c f dsebt sebg kombs ler dr f f f. (Fzo Lds 988:65-66) Cotoh: Selesk persm dferesl homoge (.)

34 re persm sederh kt dpt lgsg megtegrlk tk memperoleh peeles. Deg megtegrlk persm (.) mert kt peroleh c (.) Dm c dlh st kostt pegtegrl. Deg megtegrlk persm (.) mert kt dptk c (.) c Dm c dlh kostt pegtegrl l. Tp meghrk l-l g dberk kepd kostt-kostt c d c pert tk g dberk oleh (.) dlh st peeles persm dferesl (.). perhtk bhw d dlh peeles persm (.) d persm (.) mejkk bhw peeles-peeles t berper khss dlm setp peeles t dpt dtk sebg kombs ler dr ked peeles t. Jd jk st peeles persm (.) mk d kostt-kostt c d c sehgg c c. Tet sj jk mk c d c d jk mk c d c. Defs. St kmpl m fgs f f f L m msg-msg terdefs d kot pd selg b dktk tergtg ler (tdk bebs ler) pd selg b. Jk d kostt-kostt c c cm g tdk sem bersm-sm deg ol sehgg

35 c f( m m ) c f ( ) c f ( ) (.4) tk setp dlm selg b. Dlm hl l fgs-fgs t dsebt bebs ler dlm selg t. (Fzo Lds 988:66) Cotoh: Tjkk bhw fgs-fgs f ( ) ( / 5) d f ( ) 5 4 tergtg ler pd selg ( ) b Seblk bktk bhw fgs-fgs ler pd selg Peeles: f ( ) d f ( ) bebs Utk mejkk bhw fgs-fgs f d f tergtg ler kt hrs mejkk d kostt c d c tdk ked-d ol sehgg c c (5 4) 5 tk setp dlm selg. I dmgkk jk kt plh msl c 5 d c -. b t mslk f d f tergtg ler pd selg. Yt d kostt c d c tdk ked-d ol sedemk hgg c c tk setp dlm selg. Tetp tk d ddptk c c d c c.

36 St-st peeles sstem dlh c c. I berlw deg pemsl wl bhw f d f tergtg ler. Jd f d f bebs ler pd selg. Defs. Mslk f f L f kmpl bh fgs g sem d tr-tr smp deg tr ke - kot pd selg b. Wrosk dr f f L f dhtg pd dtk oleh ( f f f ) W ; d dtetk sebg determ ( f f f ) W ; f f f f M f f f f M L M f f f f M Tp fgs g mcl dlm determ dhtg pd. (Fzo Lds988:67) Cotoh: Dketh f ( ) d f ( ) cos cr W ( f f ; ) Peeles: Dr defs d ts d fgs-fgs g dketh dpt dhtg: W Teorem. cos ( cos ; ) s cos s Jk setp fgs L m merpk peeles-peeles persm dferesl homoge g sm

37 ( ) ( ) ( ) ( ) ( ) ( ) ( ) Mk tk setp plh kostt-kostt c c L cm kombs ler c c c m m merpk st peeles jg. Bkt: D Bkt tergtg pd d sft per ( f f f ) f f f k k ( ) cf cf d m c st kostt. D s dsjk secr terperc tk m d. Adk bhw d merpk peeles persm dferesl ( ) ( ) ( ) ( ) Deg mesbsttsk c c ke dlm persm dferesl k dperoleh: ( )( c c ) ( )( c c ) ( )( c c ) ( )( c c ) ( )( c c ) ( )( c c ) ( )( c c ) ( )( c c ) c? [ ( ) ( ) ( ) ( ) ] c [ ( ) ( ) ( ) ( ) ]? () c () c?? Mslk dber fgs-fgs L g msg-msg merpk peeles persm dferesl ler homoge orde- d s

38 g dtetk pkh fgs-fgs t tergtg ler t tdk fgsfgs tergtg ler jk kt dpt memperoleh bh kostt c c L c g tdk sem ol sehgg c c c (.5) re tp merpk peeles persm dferesl dts tr-tr L d. Mk Persm (.5) dpt dtrk kl tk memperoleh sstem persm: c c c c c c c c c (.6).. c c c Persm (.6) merpk sstem persm ljbr g ler d homoge tk c c L c g tdk dketh. Sstem persm t memp peeles sel peeles trvl c c c jk d h jk determ koefse sm deg ol. Jd fgs-fgs L tergtg ler jk W ( ; ) selg defs persm dferesl t. (Fzo Lds 988:68) tk setp dlm Dpt dbktk bhw keblk jg ber: jk fgs-fgs t tergtg ler mk Wrosk sm deg ol tk st. kre t

39 terdpt ketet berkt tk megj kebebs ler (t ketergtg ler). Teorem.4 L merpk bh peeles persm dferesl (.) g ddefsk pd selg b bebs ler jk d h jk W ( ; ) tk setp dlm selg b. (Fzo Lds988:69) Cotoh: ( ) cos d ( ) s merpk peeles persm dferesl g bebs ler kre Wrosk cos s cos s s cos tdk perh ol. Sel t jg e d e merpk peeles persm dferesl g bebs ler kre Wrosk e e e e e e jg tdk perh ol. Teorem.5 Jk L merpk peeles-peeles persm dferesl (.) d m tp () ddefsk d kot pd b d ( ) pd b mk bk W ( ; ) detk ol pd b t W ( ; ) tdk perh ol pd b.

40 Bkt: Bkt tk kss. Utk sgkt kt tk ( ; ) p( ) ( ) / ( ) W d q ) ( ) / ( ). Igt bhw p d q ( ked dlh fgs g kot kre ( ). Sekrg W Jd W ( ) ( p q ) p( ) ( p ) pw sds s Deg meggk persm ( ) e b( s) e ( t) dt deg ( ) p( ) d b ( ) kt dptk bhw persm dferesl W pw memp peeles ds W ( ) W ( )ep p( s) d (.7) kre fgs ekspoe tdk perh ol mert persm d ts jk W ol pd ttk pd selg b mk W detk ol d jk W tdk ol pd ttk mk W tdk perh ol. (Fzo Lds988:7) Defs. Mslk bhw L merpk bh peeles persm dferesl (.). Mslk jg bhw fgs-fgs t bebs ler pd selg

41 defs persm dferesl. Dktk bhw fgs-fgs t membetk hmp fdmetl (t sstem fdmetl) peeles persm dferesl tersebt. Cotoh: Fgs cos d s merpk st hmp fdmetl peeles persm dferesl. Jg fgs e d e membetk st hmp fdmetl peeles persm dferesl. Determ Wrosk dgk tertm sebg pegj tk meetk pkh st hmp peeles st persm dferesl bebs ler t tdk. (Fzo Lds988:7).6 Mslh Nl Awl Defs.4: Mslh l wl persm dferesl ler orde- d d F drtk sebg: crlh sebh peeles d d persm dferesl dlm tervl < < b sedemk hgg d ttk berlk kods wl ( d d M d d ( ( ) ) ( ) ( ) )

42 deg dlm selg (smh 989) < < b d merpk kostt. Jk persm dferesl orde st mk kods wl h berp persm tggl ( ). Sedgk tk persm dferesl d orde d kods wl dlm betk: ( ) ; ( ) d m d terdpt d persm. Istlh kods wl bersl dr mekk. Dlm mekk ( ) d metk keddk sebh obek pd st d ( ) metk d kecept obek pd st. (smh989).7 Metode Vrs Prmeter Metode vrs prmeter dgk tk meetk sols khss dr persm dferesl ler tk homoge orde-. Metode dpt dgk tk meetk sols khss dr persm dferesl ler tk homoge g koefse kostt t fgs dlm st vrbel tertet deg srt peeles persm homoge g korespodes telh dketh (Fzo Lds988:) Teorem.6 Betk persm dferesl ler o homoge orde dlh ( ) ( ) ( ) ( ) ( ) b( ) (.8)

43 Deg. Mslk peeles bebs ler dr persm homoge (.8) deg ( ) b mk peeles persm dferesl (.8) berbetk ( ) ( ) d b W W p Bkt: Ak dcr fgs-fgs gr persm p (.9) dlh peeles persm dferesl (.9) Per dr perm (.9) mejd ( ) ( ) p jk dmbl ( ) mk p per p memberk ( ) ( ) p d jk dmbl ( ) mk p proses dljtk smp per ke- dr p sehgg meghslk ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p

44 d jk dmbl ( ) ( ) ( ) ( ) mk ( ) ( ) ( ) ( ) ( ) p jk dmbl memeh sstem persm ( ) ( ) ( ) ( ) b M (.) mk persm (.9) dlh peeles persm dferesl (.8). re bebs ler mk ( ) W. Msl ( ) ( ) ( ) A M L M M ; X M ; ( ) b B M B AX mk B A X ( )B A dj A det ( ) ( ) b W M M M M M

45 ( ) ( ) b W M M sehgg ( ) ( ) b W ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) W b M M M M M t ( ) ( ) b W W ; (.) deg W dlh determ g memp kolom dgt deg ( ) t ( ) ( ) ( ) ( ) ( ) W M M M M M Jd dr (.) dperoleh ( ) ( ) d b W W p (Boce d Dprm :7-9)

46 Teorem.7 Jk p merpk seles khss dr (.) seles fdmetl dr persm dferesl ler homoge g berses deg (.) t (.) d c c c kostt mk Y p c c jg merpk seles (.). (Ber 96: 6) Bkt: Mslk dketh persm dferesl tk homoge: L [ ] ( ) ( ) f f ( ) f ( ) g( ) (.) D persm dferesl ler homoge g berses deg (.) mslk L [ ] ( ) ( ) f f ( ) f ( ) (.) re seles dr (.6) mk L[ p ] g( ) p Ml sebrg seles dr (.) mk L [ ] g( ) re (.) ler mk [ ] g( ) g( ) L p Berrt p dlh seles dr (.) sehgg ses teorem. p dpt dtls sebg c p c sehgg c c p. re sebrg seles dr (.) mk c c p merpk seles dr (.). Msl dgt deg Y mk Y dlh seles dr (.6). Seljt Y merpk seles mm dr (.).

47 .8 Metode Peeles Persm Dferesl Ler Homoge A. Peeles Persm Dferesl Ler Homoge deg oefse Vrbel dlm betk Persm Cch-Eler Persm Cch-Eler dkemkk oleh Leohrd Eler th 789 kemd dkembgk oleh Agst Cch. Peeles persm dpechk oleh Joh Beroll th 7. (smh 989: 8) Persm dlm betk: ( ) ( ) (.4) deg kostt d dmk Persm Cch-Eler. Peeles persm Cch-Eler dlkk deg jl mecr peeles g bebs ler bg persm dferesl homoge g berkorespodes deg persm g dberk. Lgkh seljt dlh deg mesbstts t e t r sehgg persm dferesl (.4) berbh mejd persm dferesl deg koefse kostt (smh 989: 9). Ds k meggk sbstts (.4) sehgg dperoleh: r r M r r ( r ) r r ( r )( r ) ( r ) r sebg sols Seljt dsbsttsk (.4) d dperoleh: r [ r( r )( r ) ( r ) r( r ) ( r ) ]

48 r re mk sehgg dperoleh: r ( r )( r ) ( r ) r( r ) ( r ) (.5) persm (.5) dsebt persm krkterstk (.4). Sebg lstrs (.5) mk dmbl beberp l. Msl mk (.5) berbetk: ( r ) r r Utk mk (.5) berbetk: ( r )( r ) r( r ) r r d seters. Sepert pd persm orde- deg koefsse kostt sols mm (.4) tergtg dr jes kr-kr persm krkterstk (.5) sebg berkt:. Akr-kr rel d berbed; r r r Seles dr (.4) dlh: r r r (.6) d seles mm dr (.4) dlh: c h r r r c c (.7) b. Akr-kr rel d kembr g berkelpt Seles dr (.4) dlh: ( ) r r r l l (.8) d seles mm dr (.4) dlh: ( ) c (.9) h r r r c l c l c. Akr-kr kompleks r α ± β g berkelpt Seles dr (.4) dlh:

49 α α α cos l l α ( β l ) s( β l ) α cos( β l ) l s( β l ) α cos( β l ) l s( β l ) (.) sedgk seles mm merpk kombs ler seles. Cotoh: Msl dketh persm dferesl ler homoge: Msl > r dlh seles. (Bdr : 6-6) Berrt r r r r( r ) r d r( r r ) dsbstts ke persm dferesl dts. r r r r r ( r r ) r( r ) r r ( r r r ) r mk r r r g kr-kr dlh r r r. Ses (.7) d (.) seles mm persm dferesl homoge dts dlh: h ( c ( l ) c s( l ) ) c cos.9 Metode Nmerk Metode rt cr sedgk merk rt gk sehgg metode merk secr hrfh berrt cr berhtg deg meggk gk-gk. Sedgk secr stlh metode merk dlh tekk g dgk tk

50 memformlsk persol mtemtk sehgg dpt dpechk deg opers perhtg t rtmetk bs (tmbh krg kl d bg) (Mr 8:5). Metode merk dsebt jg sebg ltertf dr metode ltk g merpk metode peeles persol mtemtk deg rms-rms ljbr g sdh bk t lzm. Dsebt demk kre dkl persol mtemtk slt dselesk t bhk tdk dpt dselesk secr ltk sehgg dpt dktk bhw persol mtemtk tersebt tdk memp sols ltk. Sehgg sebg ltertf persol mtemtk tersebt dselesk deg metode merk. Perbed tm tr metode merk deg metode ltk terletk pd d hl t: ) Sols deg metode merk sell berbetk gk sedgk deg metode ltk bs meghslk sols dlm betk fgs mtemtk g seljt fgs mtemtk tersebt dpt devls tk meghslk l dlm betk gk. b) Deg metode merk h dperoleh sols g meghmpr t medekt sols sejt sehgg sols merk dmk jg sols hmpr (ppromto) t sols pedekt. Ak tetp sols hmpr tersebt dpt dbt setelt g dgk. Sols hmpr tet tdk tept sm deg sols sejt sehgg d selsh tr ked d selsh tersebt dmk sebg glt (error). Sedgk deg sols ltk

51 sdh pst dhslk sols sejt g ses deg ket (Mr 8:5).. Peeles PDB secr Nmerk Secr mm problem persm dferesl sell melbtk hrg wl (l wl/tl vle) g dpt dtls sebg berkt: f ( ) ( (.) ) secr merk sols problem tersebt dlh berd dlm tervl ] [ g dbg secr tetp (eqdstce) sebk bh lgkh sehgg kr lgkh (step) g dlmbgk deg h dpt ddefsk sebg h ( h Berrt peeles merk PDB deg l wl dlh meghtg l fgs d h. Pd metode ltk l wl berfgs tk memperoleh sols g k sedgk pd metode merk l wl pd persm (.4) berfgs tk meml lelr t ters. Terdpt beberp metode merk g serg dgk tk meghtg sols PDB ml dr metode g plg dsr smp deg metode g lebh telt. Dr beberp metode g d metode g plg dsr d merpk metode g mm tk merk rms-rms sols PDB dlh metode deret Tlor. Dlm meelesk PDB deg l wl metode tersebt djbrk sebg:

52 Dberk PDB: ( ) f ( ) deg l wl ( ) Mslk ( ) deg L. dlh hmpr l d. Mk hmpr dpt dperoleh deg megrk d sektr sebg berkt: ( ) ( ( ) ( )! ) ( ( )! ( ) ( L )! )! ( ) ( ) ( ) t ( h h 6 ) ( ) h ( ) ( ) ( ) L h! ( ) ( ) Secr grs besr terdpt kelompok metode dlm meelesk PDB secr merk t: ) Metode st lgkh (oe-step) Dsebt metode st lgkh kre tk meksr l ) dbthk ( st tksr l sebelm t ). Metode g termsk metode st lgkh dlh metode deret Tlor metode Eler metode He d metode Rge tt. b) Metode bk lgkh (mlt-step) ( Pd metode perkr l ) memerlk beberp tksr l ( sebelm t ( ) ( ) ( ) L. Slh st metode bk lgkh dlh metode predctor corrector. Terdpt beberp metode

53 predctor corrector dtr dlh metode Adm-Bshforth-Molto metode Mle-Smpso d metode Hmmg. Sel t dkel jg metode He g merpk metode predctor corrector k tetp bk termsk metode bk lgkh kre tksr l ) h ddsrk ( ddsrk pd tksr ). Tj tm metode bk lgkh dlh ( meggk forms dr beberp ttk sebelm t ttk L tk meghtg tksr l ) g lebh bk (Mr 8: 79 9). (. Metode Rge-tt Peeles PDB deg metode deret Tlor tdk prkts kre metode tersebt membthk perhtg tr f ( ). D smpg t tdk sem fgs mdh dhtg tr tertm bg fgs g betk rmt. Semk tgg orde metode deret Tlor mk semk tgg tr fgs g hrs dhtg (Mr 8: 84). Sel t tk medptk hsl g lebh telt dperlk t h g kecl pdhl pegg g kecl meebbk wkt htg g lebh pjg. Oleh kre t metode Rge tt merpk ltertf dr metode deret Tlor g memberk ketelt hsl g lebh besr d tdk memerlk tr fgs (Trtmodjo : 8).

54 Metode memp sft tm t:. Metode st lgkh; tk mecr h dperlk keterg g tersed pd ttk sebelm t.. Medekt ketelt metode deret Tlor smp sk dlm p h dm l p berbed tk metode g berbed d p dsebt derjt dr metode.. Tdk memerlk perhtg tr ( ) fgs t sedr. f tetp h memerlk Metode Rge-tt memp betk mm persm: Dm ( h) ( h) (Djojodhrdjo :68) hφ (.) φ dsebt st fgs kreme g dpt dterprestsk sebg sebh slope rt-rt sepjg tervl. Fgs kreme dpt dtls dlm betk mm t: φ k k k (.) Deg setp dlh kostt d besr setp k dlh: ( ) k f (.) ( p h q k h) k f (.b) ( p h q k h q k h) k f (.c) M ( p h q k h q k h q k h) k f... (.4d)

55 Persm tersebt mejkk bhw l k memp hbg berrt. Nl k mcl dlm persm tk meghtg k g jg mcl dlm persm tk meghtg k d seters. Hbg g berrt membt metode Rge-tt dlh efse dlm htg. Ad beberp tpe mode Rge-tt g tergtg pd l g dgk. Utk g dsebt metode Rge-tt orde st φ k f Utk mk ( ) f ( )h (.4) Yg sm deg metode Eler. D dlm metode Rge-tt setelh l dtetpk kemd l p d q dcr deg memk persm (.) deg sk-sk dr deret Tlor. Seljt k dtrk metode Rge-tt orde d. Deg Metode Rge-tt orde d memp betk ( k k )h (.4) ( ) k f (.4b) ( p h q k h) k f (.4c) Nl p d q devls deg memk persm (.4) deg deret Tlor orde g memp betk: h f (.5) ( ) h f ( )

56 Deg ( ) f dpt dtetk dr hkm bert (ch rle) berkt: f f d f ( ) (.6) d Sbstts persm (.6) ke dlm persm (.5) meghslk: f f h f ( ) h (.7) D dlm metode Rge-tt dcr l p d q sedemk sehgg persm (.4) ekvle deg persm (.7). Utk t dgk deret Tlor tk megembgk persm (.4c). deret Tlor tk fgs deg d vrble memp betk: g g g ( r s) g( ) s... Deg cr tersebt persm (.4c) dpt dtls dlm betk: f f f ( p h q k h) f ( ) p h q k h ( h ) Betk dts d persm (.4b) dsbsttsk kedlm persm (.4) sehgg mejd: At h f ( ) hf ( ) q h f f ( ) ( h ) p h f ( ) ( ) f f h f f p q f h h ( ) ( ) (.8)

57 Deg membdgk persm (.7) d (.8) dpt dsmplk bhw ked persm ekvle pbl: (.9) p (.9b) q (.9c) Sstem persm dts terdr dr tg persm megdg empt blg tk dketh sehgg tdk bs dselesk. Utk t slh st blg tk dketh dtetpk d kemd d cr ketg blg l. Dggp bhw dtetpk sehgg persm (.9) smp (.9c) dpt dselesk deg meghslk: (.4) p q (.4b) Bl dplh mk p q Sehgg persm (.4) mejd k k h (.4) Deg ( ) k f (.4b)

58 ( ) k f h k h (.4c) Deg k dlh kemrg pd wl tervl d k dlh kemrg pd khr tervl. Persm (.4) dlh slh st dr rms Rge tt order d g sm deg metode he (Trtmodjo. B. ). Rms-rms Rge-tt tk order g lebh tgg dpt dtrk deg cr g sm deg pd order d. Rms Rge-tt g dpk dlm permslh ds dlh rms Rge-tt orde empt g ddefsk sebg berkt 6 ( k k k k ) h 4 (.4) Deg ( ) k f (.4) k f h hk (.4b) k f h hk (.4c) ( h ) k 4 f hk (.4d) (Chpr ). Glt Peeles secr merk st persm mtemtk h memberk l perkr g medekt l eksk (g ber) g ses deg ket. Berrt dlm peeles merk terdpt beberp

59 keslh terhdp l eksk. Terdpt tg mcm glt t glt bw glt pemblt d glt pemotog. Glt bw dlh glt dr l dt. Glt tersebt bs terjd kre kekelr dlm mel dt slh membc skl t glt kre krg pegert mege hkm-hkm fsk dr dt g dkr. Glt pemblt terjd kre tdk dperhtgk beberp gk terkhr dr st blg. Glt terjd pbl blg perkr dgk tk meggtk blg eksk. St blg dbltk pd poss ke deg membt sem gk d sebelh k dr poss tersebt ol. Sedg gk pd poss ke tersebt tdk berbh t dkk st dgt g tergtg pkh l tersebt lebh kecl t lebh besr setegh dr gk poss ke. Sebg cotoh l: dpt dbltk mejd dpt dbltk mejd 4 Sedgk glt pemotog terjd kre tdk dlkk htg ses deg prosedr mtemtk g ber. Sebg cotoh st proses tk terhgg dgt deg proses berhgg. D dlm mtemtk st fgs dpt dpresetsk dlm betk deret tk terhgg mslk: e 4 L!! 4! Nl eksk dr e dperoleh pbl sem sk deret tersebt dperhtgk. Dlm prktek slt memperhtgk sem sk pertm smp tk terhgg. Apbl h dperhtgk beberp sk pertm sj mk hsl tdk sm deg l eksk (Trtmodjo : -).

60 Ad d jes keslh hbg tr l eksk d l perkr t:. Glt bsolt dlh keslh perbed (selsh) tr l eksk d l perkr (pedekt pd l seber). Dtlsk: e Dm dlh l seber dlh pedekt pd l seber e dlh glt Ds e dlh glt bsolt t e. eslh reltf dlh tgkt keslh g dlkk deg membdgk keslh g terjd deg l eksk. e e R Deg e R glt reltf e glt bsolt l eksk Glt reltf serg dberk dlm betk perse sebg berkt: e R e % (Djojodhrjo :5)

61 . j egm ehdp zm sekrg lm pegeth d tekolog semk megs d mk tk dpt megs lm pegeth d tekolog g sempr hrs ddsr deg Al-qr d hdst sebg pegg dlm kehdp sehr-hr. re Al-qr d hdst merpk smber gm Islm g telh memberk kepd kt tetg fkt-fkt lmh g kelk dtemk d dbktk oleh eksperme Ss mt Islm. D dlm Al-Qr terdpt perth-perth g meer kepd ms tk mekk eksstes Th mell cpt-n memperhtk keks-n mell k relts g terhmpr ls d lgt d d bm. Mk kt sebg mkhlk Allh SWT hrs bs meggk p g d d lgt d d bm deg sebk-bk dlm kehdp sehrhr. Sepert g djelsk pd QS. Al-A l:- g berb sebg berkt: γsù s% Ï%!$#ρ θ sùt{ Ï%!$# Art: Yg Mecptk d meemprk (pecpt-n) D g meetk kdr (msg-msg) d member petjk. Alm semest sert segl s dcptk Allh deg kr-kr g cermt d telt deg perhtg-perhtg g mp d deg rms-rms sert persm g sembg d rp. Sem g d d lm d kr d htg-htg d rms t d persm. Seorg hl mtemtk t fsk tdk

62 membt rms tetp meemk d membolk kre Allh telh meedk. Persm dferesl dlm perkembg lm d tekolog sekrg merpk slh st mter g dpt djdk dlm pemodelpemodel mtemtk. Ms seber bk membt sest g br tetp merek h mecr pesm-persm t rms-rms g berlk pd st feome. Sepert demm berdrh tberkoloss bhk fl brg tert memp tr-tr g mtemts. Segl sest telh dcptk deg kr perhtg rms t persm tertet g rp d telt (Abdsskr 7:79-8). Dlm Al-Qr terdpt bk t-t g mejelsk berbg kosep dlm mtemtk. Ses dlm pembhs mege persm mk t g dgk t Al Bqrh t 6 berkt: Èe ä. Îû Ÿ Î/$Ζ ìö7 ômtf;/ρr& >π 6m È svϑ. «!$# È Î6 Îû óοßγs9 θøβr& tβθà)ï Ζムt Ï%!$# ã sw Β íοšîtæ ìå ρª!$#ρ â!$t±o ϑï9ß#ïè ŸÒリ!$#ρ 7π 6mèπs ($ÏΒ7s#ç7/Ψß Art: Permpm (fkh g dkelrk oleh) org-org g mefkhk hrt d jl Allh dlh serp deg sebtr beh g membhk tjh blr pd tp-tp blr serts bj. Allh melpt gdk (gjr) bg sp g D kehedk. D Allh Mh Ls (kr-n) lg Mh Megeth. Pd QS Al-Bqrh t 6 d ts mpk jels bhw Allh meetpk phl mefkhk hrt d jl Allh deg rms mtemtk. Phl mefkhk hrt dlh tjh rts kl. Secr mtemtk dperoleh persm 7

63 ( Deg metk l fkh d metk l phl g dperoleh (Abdsskr 7: 8). Sel t d ts Allh mejelsk pl sft Mh mtemtsn t dlm srt Al-Am t 6. Ÿω öνèδρ$γ WÏΒ ωî) # t øgä ŸξsùÏπ ÍhŠ 9$Î/!%` tβρ $γï9$swøβr& ç ô³tã ã&s#sùïπζ ptø:$î/!%ỳ tβ tβθßϑôàムArt:Brgsp membw ml g bk mk bg (phl) seplh kl lpt ml; d brgsp g membw perbt jht mk d tdk dber pembls melk sembg deg kejht sedg merek sedktp tdk d (drgk). Pd QS Al-Am t 6 tersebt sekl lg Allh meggk rms mtemtk tk meetk bls perbt kebk d kejht medptk bls kl ml kejht tersebt. Secr mtemtk dperoleh rms Utk ml kebk d Utk ml kejht. Vrbel metk l ml d metk l bls g dperoleh (Absskr 7: 8). Ad t l dlm Alqr g secr tersrt memerthk mt Islm tk mempeljr mtemtk k berke deg mslh frdh. Mslh frdh dlh mslh g berke deg pegtr d

64 pembg hrt wrs bg hl wrs mert bg g dtetk dlm l-qr. Sebelm dlkk pembg hrt wrs beberp hk d kewjb g bertl deg hrt wrs hrs dselesk terlebh dhl msl wst d tg. Sedgk tk pembg hrt wrs perl dketh terlebh dhl beberp jmlh sem hrt wrs g dtgglk berp jmlh hl wrs g berhk meerm d beberp bg g berhk dterm hl wrs. Berke deg bg g berhk dterm oleh hl wrs l-qr mejelk dlm srt A-Ns t d 76. Yg rt:. Allh mesrtk bgm tetg (pembg psk tk) kkm. Yt : bhg seorg k lelk sm deg bgh d org k peremp; d jk k t sem peremp lebh dr d Mk bg merek d pertg dr hrt g dtgglk; jk k peremp t seorg sj Mk memperoleh sepro hrt. d tk d org b-bp bg msg-msg seperem dr hrt g dtgglk jk g meggl t memp k; jk org g meggl tdk memp k d dwrs oleh b-bp (sj) Mk b medpt sepertg; jk g meggl t memp beberp sdr Mk b medpt seperem. (Pembgpembg tersebt d ts) sesdh dpeh wst g bt t (d) sesdh dbr htg. (Tetg) org tm d k-km km tdk megeth sp d tr merek g lebh dekt (bk) mft bgm. dlh ketetp dr Allh. Sesggh Allh Mh megeth lg Mh Bjks.. D bgm (sm-sm) seperd dr hrt g dtgglk oleh stersterm jk merek tdk memp k. jk ster-sterm t memp k Mk km medpt seperempt dr hrt g dtgglk sesdh dpeh wst g merek bt t (d) sedh dbr htg. Pr ster memperoleh seperempt hrt g km tgglk jk km tdk memp k. jk km memp k Mk Pr ster memperoleh seperdelp dr hrt g km tgglk sesdh dpeh wst g km bt t (d) sesdh dbr htg-htgm. jk seseorg mt bk lk-lk mp peremp g tdk megglk h d tdk megglk k tetp memp seorg sdr lk-lk (seb sj) t seorg sdr peremp (seb sj) Mk bg msg-msg dr ked jes sdr t seperem hrt. tetp jk sdr-sdr seb t lebh dr

65 seorg Mk merek bersekt dlm g sepertg t sesdh dpeh wst g dbt oleh t sesdh dbr htg deg tdk member mdhrt (kepd hl wrs). (Allh meetpk g demk t sebg) srt g ber-ber dr Allh d Allh Mh megeth lg Mh Pet. 76. Merek memt ftw kepdm (tetg kllh)[87]. tklh: "Allh member ftw kepdm tetg kllh (t): jk seorg meggl d d tdk memp k d memp sdr peremp Mk bg sdr g peremp t seperd dr hrt g dtgglk d sdr g lk-lk mempsk (selrh hrt sdr peremp) jk tdk memp k; tetp jk sdr peremp t d org Mk bg ked d pertg dr hrt g dtgglk oleh g meggl. d jk merek (hl wrs t terdr dr) sdr-sdr lk d peremp Mk bhg seorg sdr lk-lk sebk bhg d org sdr peremp. Allh meergk (hkm ) kepdm sp km tdk sest. d Allh Mh megeth segl sest. At dts mejelsk bg-bg g berhk dterm oleh hl wrs mslh frdh jk dlm mtemtk bs d bt persm. Msl: k lk-lk k peremp bg wrs g dterm oleh k lk-lk bg wrs g dterm oleh k peremp Persm dts merpk slh st cotoh g djelsk dlm srt A- Ns.

66 4 4 Seljt ketet bg g berhk dterm oleh hl wrs dsebt frdhl mqddrh. Terdpt em mcm frdhl mqddrh t 4 6 d 8 Mslh frdh merpk ketet Allh SWT g wjb dlksk oleh mt Islm. Berke deg mslh frdh Allh berfrm dlm srt A-Ns t d 4 sebg berkt. $γïfóss? ÏΒ Ì ôfs? ;M Ζ_ ã&ù#åzô ムã&s!θß ρ!$# ÆìÏÜムtβρ «!$# ߊρß ãm š ùï? ã&s!θß ρ!$# ÄÈ ètƒ tβρ ÞΟŠÏàèø9$# ã öθ ø9$# š Ï9 sœρ $γšïù š Ï$Î# z ã γ ΡF{$# Ñ Îγ Β ÑU# tã ã&s!ρ$γ Ïù#V$Î# z# $tρã&ù#åzô ムçνšρß ã ètgtƒρ Art:. (Hkm-hkm tersebt) t dlh ketet-ketet dr Allh. Brgsp tt kepd Allh d Rsl-N sc Allh memskk kedlm srg g meglr ddlm sgsg sedg merek kekl d dlm; d Itlh kemeg g besr. 4. D Brgsp g medrhk Allh d Rsl-N d melggr ketet-ketet-n sc Allh memskk ke dlm p erk sedg kekl d dlm; d bg sks g meghk. Bklh telh dsebtk deg tegs bhw mslh frdh dlh ketet dr Allh g wjb dlksk. Utk dpt memhm d dpt melksk mslh frdh deg bk mk hl perl dphm lebh dhl dlh kosep mtemtk. Jd d mslh frdh dpt drtk bhw mt Islm perl mempeljr mtemtk.

67 Dlm srh Al-Qdr jg terdpt lm mtemtk mege persm t É#ø9r& ô ÏΒ ö {Í ô s)ø9$# äs#ø s9 Í ô s)ø9$# äs#ø s9$tβ 7 Šr&!$tΒρ Í ô s)ø9$# Ïs#ø s9 Îûçµ oψø9t Ρr&!$ ΡÎ) 4 Lm } Ïδ íο 9 ö r& Èe ä. ÏΒ ΝÍκÍh5 ÈβøŒÎ*Î/ $pκ Ïù ßρ 9$#ρ èπsí ϑø9$# ãα t s? 9 öκ Ì ôf ø9$# ÆìôÜtΒ Art:. Sesggh m telh merk (Al Qr) pd mlm keml. D thkh km Apkh mlm keml t?. Mlm keml t lebh bk dr serb bl. 4. Pd mlm t tr mlkt-mlkt d Mlkt Jbrl deg z Th tk megtr segl rs. 5. Mlm t (peh) esejhter smp terbt fjr. Srh l-qdr dts meergk mege lltl qodr mlm keml dlm sebh hdst g drwk oleh bkhor d mslm mejeslk brg sp g berbdh pd llh l-qdr kre m d meghrpkkerdo Allh dmp dos-dos g terdhl. Hdst tersebt sdh tet medorog km mslm tk berbdh g sebk-bk pd mlm-mlm g dperc sebg lltl lqdr. Aplg kre Al-Qr meergk bhw mlm t lebh bk dr pd serb bl. Bk g berggp brg sp berbdh pd mlm t k dhtg oleh Allh sebg bdh g member phl besr selm serb bl t lebh dr 8 th. Al-Qr merpk slh st kedh mkjzt d bkt bhw ber-ber bersl dr Sg Pecpt ms d Sg Pecpt lm r.

68 Ms dhrpk dpt bekerj sebg khlfh-n d ts bm mejd cerm ctr Allh Sg Pecpt d megts segl kelemh sehgg dpt lebh dekt kepd kesempr Th. Slh st cr g dgk sebg seorg khlfh dlm megtr p g d d bm t deg meggk lm htg g dsebt jg deg lm mtemtk. re pecpt ms bertj tk mecp kesempr tertgg tergtg pd kerj kers d sh g ters-meers.

69 BAB III PEMBAHASAN Peeles persm dferesl ler tk homoge deg koefse vrbel g k dbhs pd tgs khr dlh persm dferesl ler Cch-Eler orde-. Persm Cch-Eler orde- berbetk: ( ) b (.) Deg srt wl ( ) ( ) ( ). deg d dlh kostt d. Utk lebh memhm sols ltk deg metode vrs prmeter d sols merk deg metode Rge-tt orde-4 tk meelesk persm dferesl ler Cch-Eler orde- dts mk dberk sol sebg berkt:. 4 > ( ) ( ).. 6 6l > ( ) ( ) ( ) Sols Altk Persm Dferesl Ler Cch-Eler Persm (.) merpk persm dferesl ler Cch-Eler tk homoge. Jk b ( ) mk dsebt persm Cch-Eler homoge;

70 (.) Msl dsmsk bhw merpk peeles fdmetl dr persm (.). Mk ses Teorem. seles mm dr (.) dlh: h c (.) c c Deg c c c kostt. Mert Teorem.7 seles mm st persm dferesl tk homoge berbetk ( ) h p deg h sebg seles persm dferesl ler homoge g berses deg persm dferesl tk homoge g dketh sepert (.) d p sebg seles khss pd bb k dcr seles khss dr (.) deg mecr fgs-fgs sebg peggt c c c g merpk prmeter (kostt) pd seles mm persm dferesl homoge (.). Deg demk seles khss dr (.) berbetk: p (.4) Dr (.) d (.4) dperoleh seles mm dr (.) t Y c (.5) c c Lgkh-lgkh tk mecr fgs-fgs pd (.4) sebg peggt c c c pd (.) sehgg dperoleh (.5) dsebt sebg metode vrs kostt t vrs prmeter.

71 Pd (.4) terdpt fgs g belm dketh t. Jd k dbthk persm tk meetk fgs-fgs tersebt. Dr p bs dperoleh persm-persm g k memeh (.). Deg begt k dperoleh persm-persm g dbthk deg cr merk p smp tr ke. Jk (.4) dtrk mk k dperoleh: ( ) ( ) p (.6) Mecr p g merpk seles khss dr persm dferesl homoge orde tgg sgt rmt d slt. Jl g dtemph tk megeth pemech dlh deg megsmsk d srt tet g sft mecegh mcl sk-sk g k megrg ke tr orde tgg dr. Srt pertm tk p dlh. Berrt p. Begt jg tk p dmbl srt ked t sehgg p. mk dperoleh: (.7) d jg p p p (.8)

72 Dr (.7) dperoleh persm sebg srt-srt tk mecr emd k dcr persm ke-. p merpk seles dr (.) berrt ( ) p. b (.9) (.8) dsbsttsk pd (.9) sehgg: ( ) ( ) ( ) ( ) b( ) ( ) ( ) ( ) (.) ( ) b( ) Ses fkt bhw merpk seles fdmetl dr (.) tk msg-msg berlk: Sehgg dperoleh: ( ) b (.) (.) dlh srt ke- tk mecr p.

73 Dr (.7) d (.) srt tk meetk p sebg seles khss dr (.) dlh: ( ) b (.) Dr (.) dpt dbetk: ( ) b (.) Mslk A re merpk seles bebs ler dr (.) ses Teorem.4 determ A t [ ] W tdk ol. Ses Teorem. sstem (.) dpt dselesk deg meggk tr Crmer tk mecr t: [ ]( ) ( ) ) ( b W W ; (.4) deg ( ) W dlh determ mtrks g dperoleh deg meggtk kolom ke dr A deg ( ) b.

74 Seljt [ ]( ) ( ) W b ( ) d ; (.5) W Jd seles khss dr (.) dlh: ( ) W b d p ( ) ( ) W [ ]( ) (.6) Sebelm smp pd cotoh sol pels gkpk kembl lgkhlgkh peeles secr sgkt. Secr grs besr metode vrs prmeter dpt dlkk deg lgkh-lgkh sebg berkt:. Metk persm Cch-Eler homoge orde- dlm betk. Mecr fgs komplemeter h ( ) dr persm Cch-Eler homoge deg meggk persm krkterstk ( ) c ( ) c ( ) c ( ) h dm c c c dlh kostt sebrg d { } dlh hmp fdmetl dr persm Cch-Eler homoge.. Mecr tegrl prtklr ( ) ( ) ( ) ( ) p deg cr:. Meetk Wrosk dr hmp fdmetl persm Cch- Eler [ ]( ) W b. Meetk b W ( ) W ( ) d

75 Dm W dlh determ g dperoleh deg meggt kolom ke- dr [ ]( ) deg kolom [ ] W c. Itegrl prtklr: p ( ) ( ) ( ) ( ) b W ( ) W ( ) [ ]( ) d 4. Peeles mslh l wl persm Cch-Eler tk homoge orde- dperoleh deg mmskk srt wl ( ) ( ) ( ) pd peeles mm ( ) persm Cch-Eler tk homoge dm ( ) ( ) ( ). h p Cotoh Selesk mslh l wl 4 > ( ) ( ). () Peeles: Persm homoge g berses deg () dlh 4 ()

76 Ses pejels pd sb bb.8. dlkk lgkh-lgkh berkt. Mslk r dlh seles dr () g ll dsbsttsk pd (). r ( r ) r r r 4 r r r ( r r) r 4 r ( r 4) r () Persm krkterstk dr () dlh r 4 g kr-kr dlh r d r. Ses deg (.9) seles dr () dlh ( l ) s( l ) Seles mm dr () dlh cos. h ( l ) c s( l ) c cos (4) Dr persm () dtetk trr dr h sebg berkt: h ( l ) c cos( l ) c s (5) Utk mecr l c k ddptk: c ( l) c s( l) cos Sehgg dperoleh: c c mk sbstts srt wl ( ) Dr srt wl g dberk ( ) dperoleh: c s ( l) c cos( l) Sehgg dperoleh: c ke persm (4) dsbstts ke persm (5) mk

77 emd sbstts l c d c pd persm (4): h cos ( l ) s( l ) Jd seles mm dr () dlh: h s( l ) (6) Srt tk meetk cos Mk ( l ) s( l ) s p dlh: ( l ) cos( l ) [ cos( l ) s( l ) ]( ) s( l ) cos( l ) cos ( l ) s ( l ) W W ( ) W ( ) s( l ) cos( l ) s( l ) cos( l ) s( l ) cos( l ) Ses (.5) mk ( ) ( l ) s d s( l ) d (7) Utk ( l ) s d dgk tegrl prsl

78 Dmslk: s( l ) mk d cos( l )d dv d mk v ( ) s l d v v d s( l ) cos( l ) d s( l ) cos( l ) d (8) emd tk ( l ) Msl: cos( l ) cos d dtegrlk prsl lg mk d s( l )d dv d mk v ( ) cos l d v v d cos( l ) s( l ) d cos( l ) s( l ) d (9) emd deg mesbsttsk persm (9) ke (8) dperoleh: s ( ) d s( l ) cos( l ) s( l ) d l s( l ) cos( l ) s( l ) d s l cos l s l cos l 4 4 ( s( l ) cos( l ) ) () 4 s( l ) d ( ) ( ) s ( ) d ( ) ( ) l

79 emd deg mesbsttsk persm () ke (7) dperoleh: ( ) s ( l ) cos ( l ) 4 ( ) ( s( l ) cos( l ) ) 8 Ses (.5) mk ( cos( l ) ) d cos( l ) d () Utk ( l ) cos d dgk tegrl prsl Dmslk: cos( l ) mk d s( l )d dv d mk v ( ) cos l d v v d cos( l ) s( l ) d cos( l ) s( l ) d () emd tk ( l ) Msl: s( l ) s d dtegrlk prsl lg mk d cos( l )d dv d mk v ( ) s l d v v d s( l ) cos( l ) d s( l ) cos( l ) d ()

80 emd deg mesbsttsk persm () ke () dperoleh: cos ( ) d cos( l ) s( l ) cos( l ) d l cos( l ) s( l ) cos( l ) d cos l s l cos l s l 4 4 ( s( l ) cos( l ) ) (4) 4 cos( l ) d ( ) ( ) cos ( ) d ( ) ( ) l emd deg mesbsttsk persm (4) ke () dperoleh: ( ) s ( l ) cos ( l ) 4 ( s( l ) cos( l ) ) 8 ( ) Ses Teorem.6 seles khss dr () dlh ( ) p ( s( l ) cos( l ) ) cos( l ) ( s( l ) cos( l ) ) s( l ) C 8 C 8 Dr (6) d (5) seles mm tk () dlh: 8 ( ) s( l ) C 8 (5) Cotoh Selesk mslh l wl 6 6l > (6) ( ) ( ) ( ).

81 Peeles: Persm homoge g berses deg (6) dlh (7) Ses pejels pd sb bb.8. dlkk lgkh-lgkh berkt. Mslk r dlh seles dr (7) g ll dsbsttsk pd (7). r r ( r )( r ) r( r ) r r r r r r ( r r r) ( r r) r r ( r r r ) Persm krkterstk dr (8) dlh r ( r )( r )( r ) r r r g kr-kr dlh r r d r. r (8) Ses deg (.7) seles dr (7) dlh. Seles mm dr (7) dlh h c c c (9) Dr persm (9) dtetk tr pertm d ked dr h sebg berkt: h c c c () h c c ()

82 Utk mecr l c c (9) k ddptk: c c c Sehgg dperoleh: c mk sbstts srt wl ( ) ke persm 6 9 c c c () 6 Dr srt wl g dberk ( ) dperoleh: c c c Sehgg dperoleh: dsbstts ke persm () mk 6 7 c c c () 6 Dr srt wl g dberk ( ) dperoleh: c c Sehgg dperoleh: dsbstts ke persm () mk 8 45 c c (4) 8

83 Dr persm () d () dkerjk deg meggk metode elms pbl persm () dkrg deg () mk dperoleh: c c (5) 6 Dr persm (4) d (5) dkerjk deg meggk metode elms pbl persm (4) dkrg deg (5) mk dperoleh: 44 c 4 emd tk meetk l c mk l c dsbsttsk pd persm (4) dperoleh: 44 c c Sehgg dperoleh: c emd tk meetk l c mk l c db c dsbsttsk pd persm () dperoleh: c Sehgg dperoleh: c

84 Mk deg metode elms dperoleh: c 8 47 c d c 4 emd sbstts l-l tersebt pd persm (9): 47 h Jd seles mm dr (7) dlh: h 44 4 (6) Srt tk meetk p dlh: Mk 6 6l [ ]( ) W ( ) ( 4 ) 6 W ( )

85 W ( ) W ( ) Ses (.5) mk ( ) ( 6 6l ) ( 6 ) d 6 6l d 6 l d l d d l d l C ( ) ( l ) C l C ( )( ) ( 6 6l ) ( 6 ) d l d d 4 l d l d l C

86 ( )( ) ( 6 6l ) ( 6 ) d 6 d l d l d l l d C Ses Teorem.6 seles khss dr (6) dlh 4 p ( ) l l l C 4 l l l C 4 7 l C (7) 4 Dr (6) d (7) seles mm tk (6) dlh: ( ) l C. Sols Nmerk Persm Dferesl Ler Cch-Eler Persm dferesl ler Cch-Eler (.) dts k dbhs deg meggk metode merk t metode Rge-tt orde-4. Peeles persm dferesl ler Cch-Eler tersebt meggk progrm dlm bhs Mtlb g dpt dlht pd lmpr. Secr mm lgortm t lgkh-lgkh dlm meelesk persm dferesl ler Cch-Eler deg metode Rge-tt Orde-4 dlh:

87 ) Meetk persm dferesl ler Cch-Eler g k dselesk d merbh mejd persm dferesl ler Cch-Eler tgkt st. ) Megmbl besr l wl vrbel bebs ( ) d vrbel terkt ( ) ( z ) d ( w ) ) Meetk l ( ) g k dtetk peeles besert besr h (kr lgkh). 4) Melsk formls rms metode Rge-tt orde-4. 5) Meghtg vrbel-vrbel g terdpt dlm rms deg meggk formls rms g telh dtetk t vrbel k smp k 4 d m smp m sert t smp t 4. 6) Meghtg z d w deg mesbsttsk vrbel-vrbel g telh ddptk pd lgkh 5 ke dlm formls rms metode Rge- tt Orde-4. Cotoh 4 () ( ) ( ).

88 Peeles Lgkh Dketh Persm dferesl ler Cch-Eler orde-: 4 4 f ( ) Mslk z Mk z g( ) g( z) 4 Sehgg dperoleh sstem PD orde- d d dz d ( z) z f g z ( z) 4 z Lgkh Dberk l wl ( ) ( ) z( ) d z t dpt dtls dberk l Lgkh Pels meetk g k dselesk dlh pd st deg kr lgkh h.

89 Lgkh 4 Ses deg formls rms (.5) g terdpt pd bb II d deg. h mk ( ) ( )( ) k k k k k k k k h ( ) ( )( ) m m m m m m m m h z z Lgkh 5 re. h mk ( ) z f k ( ) z g m k hm z hk h f ( ) ( ) ( )... m z k f m hm z hk h g ( ) ( ) ( )... m z k g k hm z hk h f ( ) ( ) ( )... m z k f m hm z hk h g ( ) ( ) ( )... m z k g 4 k ( ) hm z hk h f ( ) ( ) ( ) ( )... m z k f

90 m 4 g ( h hk z hm ) g( (. ) (.) k z (. ) m ) Utk ters g pertm deg z z mk ddpt: k ( z ) ( ) f f m ( z ) ( ) 4 g g k ( ) ( ) ( ) f.. k z. m f (. ) (.)( ) (.)( ) f (.5.5 ) g.. k z. m g (. ) (.)( ) (.)( ) g(.5.5 ) m ( ) ( ) ( ) k f (. ) (.) k z (. ) m f f ( ) g.. k z. m g g( ) ( ) ( )( ) ( )( ) m ( ) ( ) ( ) ( ) ( )( ) ( )( )

91 k 4 f ( (. ) (.) k z (. ) m ) f ( (. ) (.)(.99658) (.)( ) ) f ( ) g.. k z. m m 4 ( ( ) ( ) ( ) ) g ( (. ) (.)(.99658) (.)( ) ) f ( ) Lgkh 6 Berdsrk vrbel-vrbel g telh ddpt pd lgkh 5 mk besr d z dlh: (.)( k k k k ) 4 6. k k k k ( )( ) ( )( ) z z (.)( m m m m ) 4 6 z. m m m m z ( )( ) z ( )( ( ) ( ) ( ) ) D h...

92 Jd pd st. besr dlh d z dlh Iters ters berlg hgg mecp t ters ke sehgg pd khr dperoleh peeles ( ) Secr keselrh peeles merk persm dferesl ler Cch-Eler () dkerjk deg Mtlb. Cotoh 6 6l () ( ) ( ) ( ). Peeles Lgkh Dketh Persm dferesl ler Cch-Eler orde-: 6 6l Mslk z z w 6 6l w z Mk w q( ) q( z w) Sehgg dperoleh sstem PD orde- d d ( z w) z f

93 dz d ( z w) w g dw 6 6l w z ( ) q z w d Lgkh t dpt Dberk l wl ( ) ( ) z( ) ( ) w( ) dtls dberk l z d 6 45 w. 8 Lgkh Pels meetk g k dselesk dlh pd st. 5 deg kr lgkh h. 5 Lgkh 4 Ses deg formls rms (.47) g terdpt pd bb II d deg h.5 mk z h 6 ( k k k k ) (.5)( k k k k ) 4 z h 6 6 ( m m m m ) z (.5)( m m m m ) 4 w 6 ( t t t t ) w (.5)( t t t t ) w h

94 Lgkh 5 re h. 5 mk f ( z w ) g( z w ) ( z w ) k m t q k m t f h hk z hm w ht f.5.5 k z.5 m w. 5 t g h hk z hm w ht g.5.5 k z.5 m w. 5 t q h hk z hm w ht q.5.5 k z.5 m w. 5 t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f h hk z hm w ht f.5.5 k z.5 m w. 5 t g h hk z hm w ht g.5.5 k z.5 m w. 5 t q h hk z hm w ht q.5.5 k z.5 m w. 5 t k m t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k 4 f ( h hk z hm w ht ) f ( (.5) (.5) k z (.5) m w (. 5) t ) m 4 g( h hk z hm w ht ) g( (.5) (.5) k z (.5) m w (. ) t ) 5

95 t 4 q( h hk z hm w ht ) q( (.5) (.5) k z (.5) m w (. ) t ) 5 Utk ters g pertm Deg 9 6 z z 7 d 6 45 w w mk ddpt: k f ( z w ) f m g( z w ) g q z w t ( ) q l f.5.5 k z.5 m w. 5 t f (.5 ) (.5 ) (.5 ) (.5)(.5) f k ( ) ( ) ( ) ( ) ( ) g.5.5 k z.5 m w. 5 t g (.5 ) (.5 ) (.5 ) (.5)(.5) g m ( ) ( ) ( ) ( ) ( ) q.5.5 k z.5 m w. 5 t q (.5 ) (.5 ) (.5 ) (.5)(.5) q t ( ) ( ) ( ) ( ) ( ) l

96 k ( ) ( ) ( ) ( ) f.5.5 k z.5 m w. 5 t f (.5 ) (.5)( ) (.5)( ) (.5)( ) f ( ) g.5.5 k z.5 m w. 5t g (.5 ) (.5)( ) (.5)( ) (.5)( ) g m ( ) ( ) ( ) ( ) ( ) q.5.5 k z.5 m w. 5 t q (.5 ) (.5)( ) (.5)( ) (.5)( ) q t ( ) ( ) ( ) ( ) 4 ( ) 6.5 6l k f ( (.5) (.5) k z (.5) m w (. ) t ) f (.5 ) (.5)( ) (.5)( ) (.5)(.48496) f ( ) g (.5.5 k z.5 m w. 5 t ) g (.5 ) (.5)( ) (.5)( ) (.5)(.48496) g m ( ) ( ) ( ) ( ) 4 ( ) q (.5.5 k z.5 m w. 5 t ) q (.5 ) (.5)( ) (.5)( ) (.5)(.48496) q t ( ) ( ) ( ) ( ) ( ) l

97 Lgkh 6 Berdsrk vrbel-vrbel g telh ddpt pd lgkh 5 mk besr z d w dlh: (.5)( k k k k ) k k k k ( )( ) ( ) z z (.5)( m m m m ) 4 6 z.5 m m m m z ( )( ) z ( ) w w (.5)( t t t t ) 4 6 w.5 t t t t w ( )( ) w ( )( ) D h Jd pd st. 5 besr dlh.6747 z dlh d w dlh

98 Iters ters berlg hgg mecp. 5 t ters ke sehgg pd khr dperoleh peeles (.5) Secr keselrh peeles merk persm dferesl ler Cch-Eler () dkerjk deg Mtlb.. Alss Glt Persm Dferesl Ler Cch-Eler Peeles secr merk dr st persm mtemtk h memberk l perkr g medekt l eksk (ber) dr peeles ltk. Berrt dlm peeles merk tersebt terdpt keslh (glt) terhdp l eksk d ketdkpst dpt terjd. Utk meghtg l dr persm dferesl ler Cch-Eler orde- pd cotoh d tk meghtg l dr persm dferesl ler Cch-Eler orde- pd cotoh pels meggk progrm Mtlb. Hl dlkk tk meghdr ketdktelt dlm meghtg l-l tersebt pd sem g dgk. Hsl dpt dlht pd Tbel.. d Tbel.. Glt keslh dr peeles secr ltk d merk dr persm dferesl ler Cch-Eler terdr dr glt bsolt d glt reltf sepert g telh djelsk pd BAB II. Glt bsolt merpk selsh tr l eksk d l perkr. Yg dmksd l eksk d s dlh l hsl peeles g dcr secr ltk deg metode vrs prmeter. Sedgk l perkr dlh l hsl peeles g dcr deg metode merk Rge-tt orde-4. Seljt glt bsolt drmsk deg betk:

99 e D dr hsl ters dpt dhtg pl keslh relf deg betk: e R e % Hsl dr pecr glt secr keselrh dtbelk t pd tbel. d tbel. sebg berkt: Tbel.. Perbdg sols metode Rge-tt d metode vrs prmeter persm dferesl ler Cch-Eler orde- Iters Nmerk Altk % Glt Dr sols secr ltk ddpt l d hsl ters secr merk ddpt l tk besr l tr ked sols t terdpt selsh g dsebt deg glt. Jd sols ltk sebg pegotrol glt tk sols merk. eslh reltf pd ters ke-. Pd tbel glt semk tr d per l pd setp lgkh tdk terll besr sehgg tdk terll mempegrh pertmbh keslh. Sehgg keslh pd persm dferesl ler Cch-Eler stbl kre pertmbh dr keslh kecl. Hl berrt

100 bhw metode Rge-tt orde-4 merpk metode g l medekt sols eksk. Tbel.. Perbdg sols metode Rge-tt deg metode vrs prmeter persm dferesl ler Cch-Eler orde- Iters Nmerk Altk % Glt Dr sols secr ltk ddpt l d hsl ters secr merk ddpt l tk.5 besr l tr ked sols t terdpt selsh g dsebt deg glt. Jd sols ltk sebg pegotrol glt tk sols merk. eslh reltf pd ters ke-. Pd tbel glt k tetp kek setp lgkh tdk terll besr sehgg tdk terll mempegrh pertmbh keslh. Sehgg keslh pd persm dferesl ler Cch-Eler stbl kre pertmbh dr keslh kecl. Hl berrt bhw metode Rge-tt orde-4 merpk metode g l medekt sols eksk.

101 .4 Tj Agm Terhdp Hsl Pembhs Berdsrk hsl pembhs d ts bhw dlm peeles persm dferesl ler tk homoge deg koefse vrbel dpt dselesk bk secr ltk mp merk. Peeles secr ltk t deg meggk metode vrs prmeter tk mecr kekrt hsl sedgk peeles secr merk t deg meggk metode Rge-tt orde-4 memberk l perkr g medekt l eksk (ber) dr peels ltk. D dr hsl pembhs d ts persm dferesl ler tk homoge deg koefse vrbel lebh mdh d lebh cept dselesk deg metode merk dr pd deg meggk sols ltk. Sedgk d dlm gm Allh memberk kemdh bg mt tk meelesk segl mslh. Dlm hl kemdh sgt dbthk dlm meelesk persm tertm dlm bdg lm mtemtk. Sepert g terctm dlm QS Al-Bqrh:85 sebg berkt:!$# (#ρç É96çGÏ9ρ ο Ïèø9$# (#θèïϑò6çgï9ρ ô ãèø9$# ãνà6î/ ß ƒì ムŸωρ t ó ãšø9$# ãνà6î/ ª!$# ã ƒì ムšχρã äô±@ öνà6 ès9ρöνä δ $tβ 4?tã Art:...Bhws Allh meghedk kemdh d tdk meghedk keskr bgm d hedklh km meckpk blg d hedklh km megggk Allh ts petjk-n g dberk kepdm sp km berskr. emdh dlm lm mtemtk dpt memberk jl g ber tk peeles persm tersebt. Dlm meelesk lgkh-

102 lgkh hrs telt tk memperoleh hsl g tept dlm perhtg secr mtemts. Adp kemdh dlm meeles permslh t khss dlm lm mtemtk perl d pejels g berkt deg hdts drwtk oleh Bkhr sebg berkt: Dr As r. dr Roslllh sw bersbd: Hedklh km memdhk d jg memperslt hedklh km memberk kbr gembr d jg meebbk org lr. (HR. Bkhr) Berdsrk hdst d ts bhws lm mtemtk t hrs dberk deg keseg gr dpt dselesk deg mdh. Pd hdts g l rwt Mslm (64) djelsk jg bhw dsmpk deg berthp (Fchrdd 996 :56). Mslh mtemtk khss persm dferesl ler tk homoge dpt dselesk bk secr ltk mp merk. Sols ltk merpk sols kot sehgg sols dr l vrbel bebs dpt dtemk sgt krt d tept. Sedgk sols merk sols dpt dperoleh dr po-po grd terpsh pproksms keslh kwtttf hrs dkedlk deg bk tk ketelt (Lm 994:) sehgg dlm pegerj lgkh dem lgkh hrs telt d cermt. Dlm Islm sgt meekk kehrs melkk peeldk g telt d pegmt g ber terhdp fkt-fkt kokret dlm lm semest tk kemd meregk tem t tk mecp eber Hkk. Sebg ms g tdk leps dr keslh mk dlm melkk perhtg hrs deg telt tk medptk keber dlm hsl perhtg. Sepert dlm QS Mrm:94 sebg berkt:

103 #t tã öνèδ tãρ Λàι9 Áômr& ô s) 9 Art: Sesggh Allh telh meetk jmlh merek d meghtg merek deg htg g telt. At d ts dphm oleh bk lm sebg D g megeth kdr setp perstw d rc bk p g terjgk oleh mkhlk mp g tdk terjgk sepert hembs fs rc peroleh rezek d kdr tk ms k d medtg. Allhlh g megeth deg mt telt rc segl sest dr seg jmlh d kdr pjg d lebr jh d dekt tempt d wkt kdr ch d gelp sebelm sedg t ketk d st wjd d l-l sebg (Shhb :57). Al-Qr megjk ms tk meeldk meggkp kejb d rhs-n sert memerthk ms tk memftk kek lm g melmph tk kesejhter hdp. Deg demk bhws l-qr megjk ms tk meksk eksstes Th mell cpt-n meggkp rhs-rhs k relts kokret g terdpt d lgt d d bm tk dmftk bg kesejhter hdp (Rhm 7:). Slh st t deg mempeljr lm mtemtk. Dlm lm mtemtk bk memberk mft bg ms dlm hl lm htg-meghtg dlm kehdp sehr-hr (mslk perhtg lm wrs perdgg d sebg) d jg bk meemk kmt

104 dr Allh g sebelm tdk keth. Al-Qr memberk petjk tetg jl g ber mej lm pegeth sert mmp medptk kesmpl g ber berdsrk pelr d observs tetg kejb d rhs Allh. Slh st cr s Ull Albb dlh sell sdr k kehdr Th pd dr dlm segl sts d kods smbl bersh megel Allh deg klb (dzkr) sert megel lm semest deg kl (pkr) sehgg smp kepd bkt g sgt t k kegg Allh Swt dlm segl cpt-n. Deg d kemj lm pegeth sem kmt g dkrk Allh kepd ms dpt deg mdh drsk dbk kejb d rhs sehgg k membw ms lebh dekt kepd Allh mell hsl-hsl perhtg g telh dlkk dlm meelesk mslh g berkt deg lm mtemtk. re pecpt ms bertj tk mecp kesempr tertgg tergtg pd kerj kers d sh g ters-meers. Tertm dlm beljr mtemtk perl d kerj kers d ters-meers dlm meelesk permslh smp medptk hsl g tept d ber. Sebg seorg hl mtemtk ketept sert krs dlm perhtg g dlkk hrs memp hsl g seksm d krt sehgg meghslk keber g shhh. Semgt lh g dtekk dlm Al-Qr. Sepert g terctm dlm Q.S A-Ns :86 sebg berkt: $ 7ŠÅ m > ó «Èe ä. 4?tã tβ%.!$# βî) Art:...Sesggh Allh sell membt perhtg ts tp-tp sest.

105 BAB IV ESIMPULAN DAN SARAN 4. esmpl Dr r d pembhs d ts dpt dsmplk bhw. Sols ltk deg metode vrs prmeter dperoleh deg meggk lgkh-lgkh sebg berkt:. Metk persm Cch-Eler homoge orde- dlm betk b. Mecr fgs komplemeter h ( ) dr persm Cch-Eler homoge deg meggk persm krkterstk ( ) c ( ) c ( ) c ( ) dm c c c dlh kostt h sebrg d { } dlh hmp fdmetl dr persm Cch-Eler homoge. c. Mecr tegrl prtklr ( ) ( ) ( ) ( ) cr: p deg Meetk Wrosk dr hmp fdmetl persm Cch- Eler [ ]( ) W Meetk ( ) W ( ) d b W

106 Dm W dlh determ g dperoleh deg meggt kolom ke- dr [ ]( ) deg kolom [ ] Itegrl prtklr: W p ( ) ( ) ( ) ( ) b W ( ) W ( ) [ ]( ) d d. Peeles mslh l wl persm Cch-Eler tk homoge orde- dperoleh deg mmskk srt wl ( ) ( ) ( ) pd peeles mm ( ) Cch-Eler tk homoge dm ( ) ( ) ( ). h p persm. Sols merk deg metode Rge-tt orde-4 dperoleh deg meggk lgkh-lgkh sebg berkt:. Meetk persm dferesl ler Cch-Eler g k dselesk d merbh mejd persm dferesl ler Cch-Eler tgkt st. b. Megmbl besr l wl vrbel bebs ( ) d vrbel terkt ( ) ( z ) d ( w ). c. Meetk l ( ) g k dtetk peeles besert besr h (kr lgkh). d. Melsk formls rms metode Rge-tt orde-4.

107 e. Meghtg vrbel-vrbel g terdpt dlm rms deg meggk formls rms g telh dtetk t vrbel k smp k 4 d m smp m sert t smp t 4. f. Meghtg z d w deg mesbsttsk vrbel-vrbel g telh ddptk pd lgkh 5 ke dlm formls rms metode Rge-tt Orde-4.. Dr sols secr ltk d hsl ters secr merk tk persm dferesl ler Cch-Eler sm-sm medptk l tetp besr l tr ked sols t terdpt selsh g dsebt deg glt. Jd sols ltk sebg pegotrol glt tk sols merk. Besr glt pd peeles persm dferesl ler Cch-Eler k tetp kek setp lgkh tdk terll besr sehgg tdk terll mempegrh pertmbh keslh. Sehgg keslh pd persm dferesl ler Cch-Eler stbl kre pertmbh dr keslh kecl. Hl berrt bhw metode Rge-tt orde-4 merpk metode g l medekt sols eksk. 4. Sr Skrps merpk peelt deg kj ltertr tetg sols persm dferesl ler tk homoge deg koefse vrbel g dterpk pd persm dferesl ler Cch-Eler. Oleh kre t pels meghrpk gr d peelt tetg sols persm dferesl ler tk homoge deg koefse vrbel g dterpk pd persm

108 l. Msl dterpk pd persm dferesl ler Legedre d deg metode g berbed.

109 DAFTAR PUSTAA Abdsskr. 6. Ad Mtemtk dlm Al-Qr. Mlg: UIN Press. Abdsskr. 7. etk Megjr Mtemtk. Mlg: UIN Press. Ato Howrd Aljbr Ler Elemeter. Jkrt: Erlgg Arhm Mhmmd d Des At. 5. Pemrogrm Mtlb. Yogkrt: Ad Azz Abdl. 6. Modl Prktkm Metode Nmerk deg Mtlb. Mlg: UIN Bdr.. Persm Dferesl d Mtemtk Model. Mlg: UMM Press. Ber H. S. 96. Dfferetl Eqtos. Msschssets: Addso-Wesle Pblshg Comp Boce Wllm C d D Prm Rchd C.. Elemetr Dfferetl Eqtos d Bodr Vle Problem. Sevet Edto. New York: Joh Wle&Sos. Ic Chpr Steve C.. Nmercl Methods for Egeers Wth Softwre Progrmmg Applcto Forth Edto. Amerk: McGrw hll Compes Ic. Djojodhrdjo Hrjoo.. Metode Nmerk. Jkrt: PT. Grmed Pstk Utm Fchrdd d Irf Fchrdd Plh Sbd Rsl (Hds-hds Plh). Jkrt: Bm Aksr. Fzo d Lds Persm Dferesl Bs deg Peerp Moder. Jkrt: Erlgg. Goode Stephe W. 99. A Itrodcto to Dfferetl Eqto d Ler Algebr. New Jerse: Pretce Hll. smh Yh S Persm Dferesl. Jkrt: Depdkbd. Lm C. Y Appled Nmercl Methods for Prtl Dfferetl Eqto. Pretce-Hll. Ic Sgpore

110 Mrdls.. Metode Peelt St Pedekt Proposl.. Jkrt: PT Bm Aksr. Mr Rld. 8. Metode Nmerk. Bdg: Iformtk. Pmtjk & Stos. 99. Persm Dferesl bs fklts MIPA. Bdg: Isttt Tekolog Bdg. Prcell J. Ed.984. lkls d Geometr. Jkrt: Erlgg Rhm Fzlr. 7. Eskloped Ilm Dlm Al-Qr. Bdg: PT Mz Pstk. Ross Sheple L Dfferetl Eqtos. Thrd Edto. New York: Joh Wle&Sos. Ic Shhb Qrsh.. Tfsr Al-Mshbh Jkrt: Leter Ht. Sswto Rd Ags P Persm Dferesl Ler Orde Tgg. Mlg: Skrps Jr Peddk Mtemtk IIP Mlg. Ssl I Nom. 99. Dsr-dsr Metode Nmerk. Bdg: F. MIPA ITB Trtmodjo Bmbg.. Metode Nmerk Dlegkp deg Progrm ompter. Yogkrt: Bet Offset. Bsmo Setjo. Pemodel d Mtemtk Terp ( dkses pd 5 Oktober 8)

111 Lmpr. Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

112

113 Lmpr. Otpt Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

114 Lmpr. Grfk Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

115 Lmpr 4. Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

116

117 Lmpr 5. Otpt Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb

118 Lmpr 6. Grfk Progrm Peeles PDL Cch-Eler Orde- deg Metode Rge-tt Orde 4 Meggk Softwre Mtlb