信号与系统奥本海姆英文版课后答案chapter2

CHAPTER2ANSWERS21AWEHAVEKNOWTHAT1KYXNHXNTHESIGNALSXNANDHNAREASHOWINFIGURES21FROMTHISFIGURE,WECANEASILYSEETHATTHEABOVECONVOLUTIONSUMREDUCESTO11YNHXHXN2THISGIVES14224NBWEKNOWTHAT2KYNXHNXNKCOMPARINGWITHEQS211,WESEETHAT21YCWEMAYREWRITEEQS211AS1KYNXHXHNKSIMILARLY,WEMAYWRITE322KCOMPARINGTHISWITHEQS21,WESEETHAT31YN22USINGGIVENDEFINITIONFORTHESIGNALHN,WEMAYWRITE3102KHUKTHESIGNALHKISNONZEROONLYINTHERANGFROMTHISWEKNOWTHATTHESIGNALHKISNONZEROONLYINTHERAGE12NHIFWENOWSHIFTTHESIGNALHKBYNTOTHERIGHT,THENTHERESULTANTSIGNALHNKWILLBEZEROINTHERANGE93KNTHEREFORE,9,A3B23LETUSDEFINETHESIGNALS12NXUANDHWENOTETHATAND1XN12HNNOW,11112KYNXHHXKBYREPLACINGKWITHM2INTHEABOVRSUMMATION,WEOBTAIN11MYXNNUSINGTHERESULTSOFEXAMPLE21INTHETEXTBOOK,WEMAYWRITES21112234011NXNN20112HN2FIGURES2112NYU24WEKNOWTHATKNXHXHKTHESIGNALSXNANDYNAREASSHOWNINFIGURES24FROMTHISFIGURE,WESEETHATTHEABOVESUMMATIONREDUCESTOTHISGIVES345678YNXHNNXHNXHN6,7128930,YNOTHERWIS25THESIGNALYNISYNXNHNKKNHXINTHISCASE,THISSUMMATIONREDUCESTOYN90KX90KFROMTHISITISCLEARTHATYNISASUMMATIONOFSHIFTEDREPLICASOFHNSINCETHELASTREPLICASWILLBEGINATN9ANDHNISZEROFORNN,YNISZEROFORNN9USINGTHISANDTHEFACTY140,WEMAYCONCLUDETHATNCANATMOSTBE4FURTHERMORE,SINCEY45,WECANCONCLUDETHATHNHASATLEAST5NONZEROPOINTSTHEONLYVALUEOFNWHICHSATISFIESBOTHTHESECONDITIONSIS426FROMTHEGIVENINFORMATION,WEHAVEYNXNHNKKNHXU131K1NREPLACINGKBYP1,YNS261013PNUFORN0THEABOVEEQUATIONREDUCESTO,YN01P23FORN0,WEHYPOTHESIZETHATFORT031TUETXJTJKE31SUBSTITUTINGFORANDINTHEGIVENDIFFERENTIALEQUATION,TJTJTJEJ31314THISGIVES,431JTHEREFORE,T0,131TJPETYINORDERTODETERMINETHEHOMOGENEOUSSOLUTION,WEHYPOTHESIZETHATSTHATSINCETHEHOMOGENEOUSSOLUTIONHASTOSATISFYTHEFOLLOWINGDIFFERENTIALEQUATION,0TYDTHWEOBTAIN,44SESETSTTTHISIMPLIESTHATS4FORANYATHEOVERALLSOLUTIONTOTHEDIFFERENTIALEQUATIONNOWBECOMES,013J13JTHEREFOREFORT0,T0,4TJTEJTYSINCETHESYSTEMSATISFIESTHECONDITIONOFINITIALREST,FORT00/NKUNFOR1BFROMAYNUNN1UNN01KNCFORN6YN0/NK10/NKKTHEREFOREYN48/91/8,6/2,NNDTHEDESIREDCONVOLUTIONISYNKXHNX0HNX1HN1X2HN2X3HN3X4HN4HNHN1HN2HN3HN4THISISSHOWNINFIGURES221222ATHEDESIRECONVOLUTIONISYTXHTDT00TTTETHEN52010114321NYNFIGURES221YT/TTTEUTUBTHEDESIRECONVOLUTIONISYTXHTD20TD52TTHISMAYBEWRITTENASYT25202521521,1,3,360,6TTTTTTTEEDTEDETETTHEREFOREYT24102421021/,1,31/,360,6TTTTTEETETEETTCTHEDESIRECONVOLUTIONISYT,XHTD20SINHTDTHISGIVEUSYT0,12/CO,13S5,5TTTELETHTT1H23WHERET1H4/3,01TOERWISNOWYTHTXTHTXT2XTWEHAVETXT1H1441133TABDATBTTHEREFOREYT22TT2TATBXTFXTPERIODICIMPLIESYTPERIODICDETERMINE1PERIODONLYWEHAVEYT122121,4337/,TTTTDDTTTHEPERIODICOFYTIS2223YTISSKETCHEDINFIGURES223FORTHEDIFFERENTVALUESOFTTHEREFORE224AWEHAVEGIVENTHATTHEREFORE,21HN22HNSINCEHN1HN2HNWEGETHN2HN11THEREFOREH0110H13H121HH2302H32133H414114H505H2H50FORN51HNBINTHISCASEYNXNHNHNHN1225AWEMAYWRITEXNASXN|13NNOWTHEDESIRECONVOLUTIONISYNHNXN10/41/343KNKKNKKKUUBYCONSIDEREACHSUMMATIONINTHEABOVEEQUATIONSEPARATELYWEMAYSHOWTHAT0T33/23/231/21T3/2YT1YTT201313TYTT021211T1143011345T4YTTFIGURES223YN412/3,/2561/3,NNNBNOWCONSIDERTHECONVOLUTION1/4NYUUWEMAYSHOWTHAT10,3/2561/3,NNALSOCONSIDERTHECONVOLUTION24NYUUWEMAYSHOWTHAT41/,3NCLEARLY,YNOBTAINEDINTHEPREVIOUSPART1YN2226AWEHAVEX12KXNK053UTHISEVALUATESTO1YNX2N4/,NOTHERWISBNOWYN31Y1THEREFOREYN432/2/,2,0NNNOTHERWISTHEREFORE,YNUN331/NCWEHAVEUN3UN2N323YNXDFROMTHERESULTOFPARTC,WEGETYNUN31XN3/2N227THEPROOFISASFOLLOWSAYYTDXHTTHADZ228ACAUSALBECAUSEHN0FORN0UNSTABLEBECAUSE0/NDNOTCAUSALBECAUSEHN0FORNA1B3233AIFROMEXAMPLE214,WEHAVEKNOWTHAT51231TUETYTIIWESOLVETHISALONGTHELINESOFEXAMPLE214FIRSTASSUMETHATISOFTHETYPFORMFORT0THENUSINGEQP2331WEGETFORT0TKE222KTTE2T41WEKNOWTHATFORT0WEMAYHYPOTHESIZETHEHOMOGENEOUSSOLUTIONTOBEOFTHEFORMTPY41THAT2THEREFORE,FORT0TTE2ASSUMINGINITIALRESTWECANCONCLUDETHAT0FORT0THEREFORE,0AA2TY41THEN,22TUETTIIILETTHEINPUTBEASSUMETHATTHEPARTICULARSOLUTIONISOFTHEFORMX33TTYPTYP1K2TFORT0USINGEQP2331,WEGET32221KTUE2TUE1K3TUE2TUET3TE2EQUATINGTHECOEFFICIENTSOFANDONBOTHSIDES,WEGET3T21/5AND1/4NOWHYPOTHESIZINGTHAT,WEGETTPETY24T3513UTTTAE2FORT0ASSUMINGINITIALREST,0A/5/4A/5/4TYTHEREFORE,T351TE3T2TEUCLEARLY,Y2YIVFORTHEINPUTOUTPUTPAIRAND,WEMAYUSEEQP2331ANDTHEINITIALRESTTX11TCONDITIONTOWRITEFORTTTHENUSINGTYP2TTEEQP2331,WEGETFORTT2K2KK1/42TTE2TTT2WEKNOWTHATFORTT2TTTEYASSUMINGINITIALREST,WECANCONCLUDETHE0FORTT,THEREFORE,2TYA4022KTT4TETHEN,TUETYTTCLEARLY,21IIICONSIDERTHEINPUTOUTPUTPAIRWHERE0FORT0IS1T2TUETY14USINGTHEFACTTHAT1,WEGETFORT01221TTEETNOWCONSIDERASECONDSIGNAL0THEN,THECORRESPONDINGOUTPUTISXTBTY22FORT0USINGTHEFACTTHAT,WEGETFORT01TENOWCONSIDERATHIRDSIGNALNOTETHATTHEOUTPUTWILLSTILLBEFORT01213TXTX13TYTCLEARLY,FORT0THEREFORE,THESYSTEMISNOTLINEAR213TYTTYBAGAINCONSIDERANINPUTSIGNALFROMPARTA,WEKNOWTHATTHECORRESPONDINGOUTPUTFORT0WITHTUEIS112214TTEYNOW,CONSIDERANINPUTSIGNALOFTHEFORMTHENFORTT,2TTUETXTTTTAT222USINGTHEFACTTHATANDALSOASSUMINGTHATTTTHEREFORE,THESYSTEMISNOTTIMEINVARIANTCINORDERTOSHOWTHATTHESYSTEMISINCREMENTALLYLINEARWITHTHEAUXILIARYCONDITIONSPECIFIEDAS,WENEEDTOFIRST1YSHOWTHATTHESYSTEMISLINEARWITHTHEAUXILIARYCONDITIONSPECIFIEDAS01YFORANINPUTOUTPUTPAIRAND,WEMAYUSEEQP2331ANDTHEFACTTHATTOWRITEXT1YT01（S2341）112,DTFORANINPUTOUTPUTPAIRAND,WEMAYUSEEQP2331ANDTHEINITIALRESTCONDITIONTOWRITET2T2,YTX0YSCALINGEQS2341BYANDEQS2342BYANDSUMMING,WEGET121212DTTTTXTTAND30YYBYINSPECTION,ITISCLEARTHATTHEOUTPUTISWHENTHEINPUTISFURTHERMORE312YTTYT3XT12TXTTHEREFORE,THESYSTEMISLINEAR3120THEREFORE,THEOVERALLSYSTEMMAYBETREATEDASTHECASCADEOFALINEARSYSTEMWITHANADDERWHICHADDSTHERESPONSEOFTHESYSTEMTOTHEAUXILIARYCONDITIONALONEDINTHEPREVIOUSPART,WESHOWTHATTHESYSTEMISLINEARWHENINORDERTOSHOWTHATTHESYSTEMISNOTTIMEINVARIANT,0CONSIDERANINPUTOFTHEFORM21TXEUFROMPARTA,WEKNOWTHATTHECORRESPONDINGOUTPUTWILLBE14TTYAUSINGTHEFACTTHAT,WEGETFORT002214TTYENOWCONSIDERANINPUTOFTHEFORMNOTETHATCLEARLY,1/X210YTHEREFORE,FORALLTTHISIMPLIESTHATTHESYSTEMISNOTTIME321/YT2/TINVARIANTEAPROOFWHICHISVERYSIMILARTOTHEPROOFFORLINEARUSEDINPARTCMAYBEUSEDHEREWEMAYSHOWTHATTHESYSTEMISNOTTIMEINVARIANTBYUSINGTHEMETHODOUTLINEDINPARTD235ASINCETHESYSTEMISLINEAR,THERESPONDFORALLT10YBNOWLETUSFINDTHEOUTPUTWHENTHEINPUTISTHEPARTICULARSOLUTIONISOFTHEFORM2T2XTT1,PYYSUBSTITUTINGINEQP2331,WEGET2Y1NOW,INCLUDINGTHEHOMOGENEOUSSOLUTIONWHICHISOFTHEFORMWEGETTHEOVERALLSOLUTION2THYTAE,T1221TYTAESINCE,WEGET0,T122TTFORT122TYTBESINCETHETOWPIECESOFTHESOLUTIONFORINAQSS2351ANDS2352MUSTMATCHATT1,WECANDETERMINEBFROMTHE2YEQUATION21WHICHYIELDS,T434TT241AWEMAYWRITEGNXNNX3YTXTYT28XTYN1/3XNYN1/31/2XNDDDFIGURES238FIGURES2391NUNUNBNOTETHATGNXNNN1THEREFORE,FROMPARTA,WEKNOWTHATXNNN1NUSINGTHISWEMAYWRITEXNN1N2N1XNN1NN1XNN2N1N2NOWNOTETHATXNHN4N22N1NN121THEREFORE,XNHN4XNN2N12XNN1NXNNN11/2XNN1N2THISMAYBEWRITTENASXNHNXN4N24N12N12N1N1/2N11/2N2THEREFOREHN21424NN1/2/1N242WEHAVEYTXTHT50DETJTHEREFORE,Y0SIN/2J500020AIF2,THENY000BCLEARLY,OURANSWERTOPARTAISNOTUNIQUEANY2K,KANDK0WILLSUFFICE0243AWEFIRSTHAVEXTHTGTDTGHXALSO,XTHTGTDGHTXFIGURES240XTT10T220HDTGHXTHEEQUALITYISPROVEDBIWEFIRSTHAVENUNHNUN1KN2013NNOW,YNNHNN1UN2IIWEFIRSTHAVEGNHNHNUN12KNK0KN210NOW,YNUNGNUNUNN1UNTHESAMERESULTWASOBTAINEDINBOTHPARTSIANDIICNOTETHATXNHNHNXNHNHN2121ALSONOTETHATXNHNUNUN1N2NNTHEREFORE,XNHNHNNSIN8NSIN8N21244AWEHAVEXTHTDTXDTHXT1NOTETHATH0FORTTAND12121BIWEHAVEYNHNXN10NKNXHNOTETHATXK0FORNNNTHEREFORE,XKN0FORNNNNNCLEARLY,THECONVOLUTIONSUMISNOTZEROIFN3232NNNANDNNNTHEREFORE,YNISNONZEROFORNNNANDNNN312010IIWECANEASILYSHOWTHAT1YMHXCHN0FORN5DFROMTHEFIGUREITISCLEARTHATYTHTXTXT6DTX12THEREFORE,Y0X612THISIMPLIESTHATXTMUSTBEKNOWNFOR1T2ANDFORTZ1/2Z1/4341Z82THEREFOREYNAB2N14NSINCEY01Y16,WEGETA1,B2,ANDYN214N2NIIHERE20ZTHEREFOREYNA1NNBASINCEY01,Y10,WEGETA1,B1,ANDYN1NIIIONLYDIFFERENCEFROMPREVIOUSPARTISINITIALCONDITIONSSINCEY01,Y1021WEGETA1,B2ANDYN12NIVHEREZ1J/2THEREFOREYN112NNAJBJSINCEY00,Y11,WEGETA,B,AND2JJSIN/4YN255AY0X01,HNSATISFIESTHEEQUATIONN112HTHEAUXILIARYCONDITIONISH01,USINGTHEMETHODINTRODUCEDINTHEPREVIOUSPROBLEMS,WEHAVEZ1/2THEREFOREHNAUSING12NTHEAUXILIARYCONDITION,HNUN12NBFORMFIGUREP255B,WEKNOWTHATIFXNN,THENWNNUNAHNTHISIMPLIESTHATYNHNUN2UN11N1NCPLUGGINGEQP2553INTOEQP2551GIVES112212NMNMMMNHXHNXXXXTHISIMPLIESTHATEQP2553SATISFIESEQP2551DIGIVENTHAT0ANDTHATTHESYSTEMOBEYSINITIALREST,WEGETA001AYATHEHOMOGENEOUSEQUATIONIS0NKHNWITHTHEINITIALCONDITIONS01,1HAIIWEHAVE10MKNBHWHEREISASABOVE1HNEFORNM0NKAWITH0,HYMYFIWEGET1,00NEVHODRIIWEGET,2ANIIIWEGET,01,4HEVLSIVWEGETCO/63IN/2N256AINTHISCASE,S20WHICHIMPLIESTHAT2TYTXAESINCEY01,A1ANDTHUNOWCONSIDEREQP256122TDLHSXDTXDEXRTHISIMPLIESTHATYTDOESSOLVETHEDIFFERENCEEQUATIONBTAKELYTAUTTHEN0NKLKTINTEGRATINGBETWEENT0ANDT0ANDMATCHINGCOEFFICIENTS,WEGET0EXCEPT1/THISIMPLIESTHATFOR0T0LANA/NYTUTAND200NYYAND10/NDTACTHEIMPLIESRESPONSEIS0KMBHTTDITAKINGRYTAUTWEGET2132RRRAUTTTTHISIMPLIESTHATANDTHEREFOREH00ANDH01CONSTITUTETHEINITIALCONDITIONSNOWMAX2R2130,1SSTHEREFORE2TTHTAEBAPPLYINGINITIALCONDITIONSWEGETA1B1THEREFORE1TTUIITHEINITIALCONDITIONSAREH00ANDH01ALSISTHEREFORESJINTHETEFROMPARTCIF,THENWILLCONTAINSINGULARITYTERMSATT0THISIMPLIESTHATMN0KMBKDT1RTAUTFINOW11023RRAUTUTTHEREFOREALSOMX0001023ATTTHISGIVESTHEINITIALCONDITIONISH05AND3152255TTHTEUEUIIHERETHEREFOREH013ANDH044AND0A3A24310118TTTUT257AREALIZINGTHATWEMAYELIMINATETHESEFROMTHETWOGIVENDIFFERENCEEQUATIONTHISWOULDGIVEUS21XNY220YNABXNTHISISTHESAMEASTHEOVERALLDIFFERENCEEQUATIONBTHEFIGURESCORRESPONDINGTOTHEREMAININGPATSOFTHISPROBLEMARESHOWNINTHEFIGURES257258AREALIZINGTHATWEMAYELIMINATETHESEFROMTHETWOGIVENDIFFERENCEEQUATIONTHISWOULDGIVEUS21XNY2211354NYNXNTHISISTHESAMEASTHEOVERALLDIFFERENCEEQUATIONBTHEFIGURESCORRESPONDINGTOTHEREMAININGPATSOFTHISPROBLEMARESHOWNINTHEFIGURES258259AINTEGRATINGTHEGIVENDIFFERENTIALEQUATIONONCEANDSIMPLIFYING,WEGET00111TTABYTYDXTATHEREFOREA/B/C/0A1B101BREALIZINGTHATWEMAYELIMINATETHESEFROMTHETWOGIVENDIFFERENCEEQUATIONTHISWOULDGIVEUS2XTY2211TTYADBXCT260AINTEGRATINGTHEGIVENDIFFERENTIALEQUATIONONCEANDSIMPLIFYINGWEGET001222221TTTTABABBYTYDDXDXTAAATHEREFORE,A,B,C,D,E12A/02A/21/AB12/02/ABBREALIZINGTHATX2TY1TWEMAYELIMINATETHESEFROMTHETWOGIVENINTEGRALEQUATIONSCTHEFIGURESCORRESPONDINGTOTHEREMAININGPARTSOFTHISPROBLEMARESHOWNINFIGURES260261AIFROMKICKOFFSVOLTAGEMUSTEQUALTHESUMOFTHEVOLTAGESACROSSTHEINDUCTORANDCAPACITORTHEREFORE,2DYTXTLCUSINGTHEVALUESOFLANDCWEGET2DYTXTIIUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFTHEDIFFERENTIALEQUATION1/2XN5/2000DDDDYN1/20DDDYNXN1/2DDDDDDDYNXN1/25/21/2BCDBDDDDDDDXNYN1/21/25/2DDDDXNYN5/21/21/2EFXNYNBCBYNAXNCYNBCAXNDAYNXNBCEYNXNACBFFIGURES259212DYTTAYBXTWILLHAVETERMSOFTHEFORMWHERESANDSAREROOTSOFEQUATION12STSTKE0ITISASSUMEDHERETHATINTHISPROBLEM,0AND1THEREFORE,THEROOTSOFTHEEQUATIONAREJANDJ0112A0S1THEHOMOGENEOUSSOLUTIONIS2JTJTHYEAND,12IIIIFTHEVOLTAGEANDCURRENTARERESTRICTEDTOBEREAL,THENKTHEREFORE,1K22COSIN/2HYTKTTBIFROMKICKOFFSVOLTAGELAW,WEKNOWTHATTHEINPUTVOLTAGEMUSTEQUALTHESUMOFTHEVOLTAGEACROSSTHERESISTORANDCAPACITORTHEREFORE,XTRCYTDYTUSINGTHEVALUESOFR,LANDCWEGETDYTXTIITHENATURALRESPONSEOFSYSTEMISHOMOGENEOUSSOLUTIONOFTHEABOVEDIFFERENTIALEQUATIONUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFDIFFERENTIALEQUATIONT1AYTBXWILLHAVETERMSOFTHEFORMWHEREISTHEROOTOFTHEEQUATION0STAE0SS01INTHISPROBLEM,1THEREFORE,THEROOTSOFTHEEQUATIONARE1THEHOMOGENEOUSSOLUTIONIS10STHYKAND,A1CIFROMKICKOFFSVOLTAGELAW,WEKNOWTHATTHEINPUTVOLTAGEMUSTEQUALTHESUMOFTHEVOLTAGEACROSSTHERESISTORANDCAPACITORTHEREFORE,2DTYTXTLCRUSINGTHEVALUESOFR,LANDCWEGET25YTXTTUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFTHEDIFFERENTIALEQUATIONFIGURES260XNYNABCDEEXNYNCDEBYNXNABCYNXNDEACBDXNYNABCDEF212DYTTAYBXTWILLHAVETERMSOFTHEFORMWHERESANDSAREROOTSOFEQUATION0112SSTKE212SITISASSUMEDHERETHATINTHISPROBLEM,2AND5THEREFORE,THEROOTSOFTHE01A2AEQUATIONAREJANDJTHEHOMOGENEOUSSOLUTIONIS0S221TJTJHYTEEAND,A1IIIIFTHEVOLTAGEANDCURRENTARERESTRICTEDTOBEREAL,THENKTHEREFORE,1K22COSSIN/TTHTK262ATHEFORCEXTMUSTEQUALTHESUMOFFORCEREQUIRETODISPLAYTHEMASSANDTHEFORCEREQUIREDTOSTRETCHTHESPRINGTHEREFORE,2DYXTMKTTSUBSTITUTINGTHEVALUESOFMANDK,WEGET24XTUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFTHEDIFFERENTIALEQUATION212DYTAYBTTWILLHAVETERMSOFTHEFORMWHERESANDSAREROOTSOFEQUATION0112SSTE212SITISASSUMEDHERETHATINTHISPROBLEM,0AND4THEREFORE,THEROOTSOFTHE01A2AEQUATIONARE2JAND2JTHEHOMOGENEOUSSOLUTIONIS01S22JTJTHYTKEASSUMINGTHATYTISREAL,WEHAVETHEREFORE,1K22COS2THYTCLEARLY,ISPERIODICHTBTHEFORCEXTMUSTEQUALTHESUMOFFORCEREQUIRETODISPLAYTHEMASSANDTHEFORCEREQUIREDTOSTRETCHTHESPRINGTHEREFORE,DTXMBSUBSTITUTINGTHEVALUESOFMANDK,WEGET10YTXTUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFTHEDIFFERENTIALEQUATIONDTABTWILLHAVETERMSOFTHEFORMWHEREISTHEROOTOFEQUATION0STAE0S01INTHISPROBLEM,1/10000THEREFORE,THEROOTOFTHEEQUATIONARETHEHOMOGENEOUSSOLUTIONIS0S41410THYTKECLEARLY,DECREASESWITHINCREASINGTHYTCIWEKNOWTHATTHEINPUTFORCEXTFORCEREQUIREDTODISPLACESPRINGBYYTTHEREFORE,2DTYTXTMBUSINGTHEVALUESOFM,BANDKWEGET2YTTXTIIUSINGTHERESULTSOFPROBLEM253,WEKNOWTHATTHEHOMOGENEOUSSOLUTIONOFTHEDIFFERENTIALEQUATION2121DTTAYBXTWILLHAVETERMSOFTHEFORMWHERESANDSAREROOTSOFEQUATION02STSKE212SITISASSUMEDHERETHATINTHISPROBLEM,2AND2THEREFORE,THEROOTSOFTHEEQUATIONARE1JAND0S11A20S1S1JTHEHOMOGENEOUSSOLUTIONIS12JTJTHYTKEASSUMINGTHATYTISREAL,WEHAVETHEREFORE,1K2360NPNAD263（A）WEHAVEYNAMTBORROWEDAMTPAIDCOMPOUNDEDAMTFROMPREVMONTH100000N101YN1DUN1THEREFORE,YN1NHPYUSINGTHEINITIALCONDITIONY0100000,WEHAVEA100000100DTHEREFORE,YN100000100D100D0NCWEHAVEY3600P100D100D361THEREFORE,D102860DTOTALPAYMENT370,2960NKETHETOUGHESTQUESTIONINTHISBOOK264AWEHAVEYTXTHTANDXTYTGTTHEREFORE,GTHTTNOW,HTGT|TNT0NKHGTNTTHEREFOREWEWANT01,2,3NKTHEREFORE,0HHGGBINTHISCASE,G01,G11/2,G21/23,ANDSOONTHISIMPLIESTHAT12NKTTTCIHERE,HT0KIIIF01,THENHTISNOTABSOLUTELYINTEGRABLEMAKINGTHESYSTEMUNSTABLEIIIHEREGT1TTTHEINVERSESYSTEMISASSHOWNINTHEFIGUREBELOWYTXTFIGURES264DIFX1NN,YNHNIFX2N1/2NN,YNHN265ATHEAUTOCORRELATIONSEQUENCESAREASSHOWNINFIGURES265BTHEAUTOCORRELATIONSEQUENCESAREASSHOWNINFIGURES265CWEGETKXYNHXNKTHEREFORE,MAYBEVIEWEDASYDELAYTX2X2N141202N1111024NX1X2N1331242202NX1X2N121505NX2X3NA1343101234NX1X3N0123NX1X4N12321202NX1X1NALSOKXYNXHKTHEREFOREMAYBEVIEWEDASYNZNYDANDAREASSHOWNINFIGURES265X266ATHEPLOTOFX1TISASSHOWNINFIGURES265BTHEPLOTOFX2TANDX2TAREASSHOWNINFIGURES265CX1TH2TX2TH3TX1TH3T0FORT4267ATHEAUTOCORRELATIONFUNCTIONARE312,014,TTXANDXTTAND7,23,42567,0TTXANDXTTTBIFTHEIMPULSERESPONSEISHTXTTTHENYTTTCWEHAVE01/21/2TYXHDMTTHEREFORE,YTISATMOST/1/20TTIFWENOWCHOOSE20THTXXTDTHEN1/1/2YCLEARLY,YTISMAXIMIZEDFORTHEABOVECHOICEOFHTDITHERESPONSESAREASSKETCHEDINFIGURES267X3X4N654101N2NX2X4N111101N1X2X3NB0123N1367631XYN432101234N1410161