MFNWtKUb<ob<R=MDLCdNVZZJ:tN>T:\\WmDqiCB`N\\@Nd\\Qgqxx`JFmodpsqaTOuToexPpWL\\NfHSJ\\RB<K=\\k>uMU]M>IL`Iva<NBYJV@PH@nb`kI]UA=YFmm]\\KrhsAAy>tVV=tS<VRaJY`qk\\kbYl?mqS`R;HJk<yJujZ<N^<lHMQwxXQQxhywyuwyqxsxnHiudEwhPSyyYYyTcQsgXXIiQeiQyyusyOaTTo=n>LJ;@RZ<LB\\J<DjZmMJhl`<LB\\:=>LR:]J<:LZ[>^<Fj:;@BCBK;C>KJ;JBFZK>[>N:J<nbQ``;@bZ:=NbuUFeSIAoSPTSeLtPXd=YaAktiuBaSZyxwXYmLWcqMu=o_dYwLR:]:>>NjtI`xFagpgrytYXtqidug__xqUopGVtqYdI`e[PyDhudVmCQoDgpLI`oO\\<N^J>[>N:BmacgUru[t>ar^OIZuewgThsHaiDkYygwgboyxqYLqV:Ux;Kxkuy`yvVyEWwiaogXES^YvfgHiwCVydeAFIQC[;F<:Kjs]NXtus=WmxyyYXxATchTuywE]r]IlYqqDutHuQtEkdtleLweTMxmPAmQvFyrvvy`]PItg?^>^bZZ<ZbG^:o^@yn`HceHxYxoH?oenwr?bgha\\AbyyxMv^dPwdXqX^niymwwtCQoixnUgi[YiI`tJx[vqx`XbpYwrqxJNZ\\><BJ;C>CDZUXcgF=SBdSeEQdIiRk?t_OCsaY\\WyqwcF?dPYvVwIv;Tw[isUCWUYZIgaEc>?b^=v=wy=sTssERGU;_fj;dhuX[sE<=R>:\\Jk<YjyTiTl]yxwqmIlJ=lvJuJfIq>eMtxwJPMdywE`qYaNT\\S<]SZQMMXLxYyItYtplFappuLBDJK:BK;y]YfPEgGyGhCxCsBjayeWiB_dZYxjqVQ_cvGX?AV:?GjwiAGSvKd:GsAiUYCImWV<YwDCBK;[>:Ke:;D:;eyCw`;TviE^SBxeEjcgtQy=scMob^arOYgNQwwqdLCyJiScOtZQwmutdwxYOYFqWAmITqc:_sPOSd]XvMTOOs_?V[SXWMgu?TZsbVoS?ggh=gQAbm;gliUWEcMawHgxXiUjWCoaXdYWHwUearmMu:kH^GyV[wAUXugUU=CtSEEGWXQC>Kj@EpQQtw<wmqr=pxKUwHMUG]n=hMDYtdeaJQi:`rvyxyQq`HgiaerFreI^ugj:@g:QjAFnfxaYiodHmVau\\?wfNjMY`hIkIhmDObp_wTfd?>lbPbi?nX?omQt;Ij:gysfc`PePW`OPgZfcuHZ=N]<Q[:HjgocNpcfXd]^v]XoEi^EIqFg`thudH]OooN?gVwcuVbUfgNYuW?cfas`>dgwaYNpSacUoxa_uVh`YirFhukouD^Z<FbmFjqO_B@sanus_asVmYOxIx[NibxNm_agHxxwYnIpyVgynIlA?wyy\\gXeaplhfed^Z>ag^^wphl`Pcdgu_vwE^ypPh_Q_ogwoxi]pfbw`hXxB>[HFl;qw[yqwax[gxoX]i_xixuxWygQg\\`wXPesV`uVcygsnvpXqhAP]YHmbWstgwVV]QgllgwcXmf>nnOfWPw?VlIX_yQxMQwsxvp`r`xrRpZHHapA]oqpqxmhPiuQiOAu>aldi_PgdKI^\\NjHYrjAsbfq=A`QipSyuPOZu_uVhefWxP?^J>[TgZrIjvymI_m]poyylGGp`>efHx:HuPWp\\`eKIZ@x`vgdWWx_yoLyyI>uiweoOucwnCFZcHgbHmSOtJ^yZquiytvqZ;p_UhsdPkq@`pO]jX\\Tw`UpkSIcY?_:?c;@^mhx_x\\OqLWGjuGmmBoIFewCQUvMWtKOedoXgWT]?sNMSCWtlCS;yuumwxWUW_ipegtuu_sEJUDMUtneUB=SaEb?oxgYupoB`Yr;ygy[cnWVgAVyWuBaeWieaig:=FJ[hD;t<WRUKB\\ErHSwt;Xwihumxdoy\\uhdiwgiWdYhS_YVwY`yeJ_g;gcIIrwkdogSUahGCB;GILKVdsIpAw]mtYyevMxCMUFIVt;huOuZsV=]WmQbuKggMtYefDcwryWRAh;eIrAbEawk;iyqvtwsMGWKuFPIgK]TmgxQICg=i_XUNyvy@x?DN[dMB\\sA\\koUXx\\YpQme]T_PUcUysTQe=WQxLMaqO>;Ug[MIiWe:gDJ[HP[xwOToivb_MPYNlDwveRqdWTanxIwpYYNyLytxy\\tsuvwyPDYX^uthdpaQt[DNZlNjXPQytQiMjtU;qojtON@pYUQnAxq<oSHuluRL<LeQjXHmuplCaVttoHetl]viimRun`mpAlriyJy<y[TQ[pY\\yQb]SmAy@<PpIp]Ex;xxyxUGesHQK;DwnpyIatHIX>DMRhOQPQd`wEhWXQK>LJ;Un:DYRHOOHwZMrVmPAuNXxKiAu;uqw`Q[utXTR]LlN\\MpyLRYqqTn@pj@Dsi]LIQpsllYaYmiOdar]qkPuLV=X;xOmtxoaq`pNWar>aJS=MOTk?irC=RRINlIxAqvnxMRImu\\wjXrvEJGTq=AV<XpWLwALLA]SVIJldoHMQsiOdiOiEy]dkRhl<uX]tLv`wSQulErFUj?AXNIOUHmgAotdPLykpTTu\\wt<u><TRhqVDX>IL^uksIwIeYDyqHHxHHXyqvMlSF]VPTNbuWZ<w<hkR`O>LXHAyFaTeHlmmpMuu?MJ;@nm<x?xwwLVwpKAhWraSvYymtOtyn>Xw:IyTtnYdyTIYSULsauHTxsyVVTYJDlFtxwMm_xXdtomAKbTPodP_pjwTXV=lH@P<HVTHV?EJIiPCQsmAyayX]lQcDUk<UZtQLmK^UJLExjqnAitumuwmVbqjwywqpq\\pYruuI<s>\\yDXOh@TAiMq]u\\aNZ<td]LlEMYYSYPMwArO]prIpM\\Npuj[pQ<UNAijLTVu]WfavkUvmTQ_UXB`n[yS]Qx<@X;HjRdlQlwgQTkqxMYJwdQ>QmaePaDT^yQJYTF`Ni=WylLlUlNxrfxMFUPnENSdW=Hk?XygewFewiayXEX`IVGXPMdQyPS_QRSMyDhucXw>ys:xUJ\\tJ`uaexNik;ItB@Q<<Ltyo^ILY=m_@VQQnKAk?elDerd<jPdmveSgmJQywZXjHLTY<qnQt:XpiqjY\\WZDvi`Uc`ndYQnxS[XOQLuiESUijAiXRmUppq;AmCyX_UToHNIhutUyrDSwLtFixjPX?ilCxv=ms\\HQZLlXmmUxKtUU\\QUOYN>LYuHXuqsKLjTqXohpW]tvhOeUxilWHQvXQYXiRGAmqXxwtV?LJ;ay\\LsSTyMlK;DRbQUU=uT=QcUvZ<rsAJHYxjTsCxv=@J[eU>Dv:YYSiUqYoXYQtxj<al]iY?xuYUpFEtS<NMUtMMonTU\\dK=LjN@rQIqLmuYpq]IQJqVjpj>\\N=\\n<lQ[TStQyvMV=ElodpGlSVuMY`X=LjjHUf\\Vg\\qYExvynXPYDeQyQviYNeAYgitYXTqpumyOELQPLu=UJ\\aX>mQQxNgtJ]aj]ykahM_=W<=ufTwCxrYdTalmjMjMAy_Uo:hr[@TsulHlVWMyq=lVpOiutx`vf<l]Xvm\\pJHn]asvQvlQrw=UjINZMNiHxcPLgqxUQx\\XYn@p<YX<IS`<puLs<uR;YMbend=qc\\JpurKqWodVcps?hOHQY<`jg`Q`iyrmNApQ@@x:Dq=lV:=sQuuk=rw=ywXOMeK>PSpHlDQX_mtqxUVXOYmNPWZ;yjUqm>Ffiha`?^NaleFpoWbXVnMXl;XgB@`?PidiiT_h>N^NxZ<P\\xwb>Qnxqu\\gu<gkGW^Nnt@?_qNuOVvfi_ewtLFwJi]MYnBq`qi]Ihlvxx<NoPxfWpae@eui_fhe`^]lfoUH[KAaBQqwgjv_sv`xsGjxHtaArnaZtif`^pY_sl?wmO_Gw]aFwYo_Uvka_soO[Afww`nM`scN]]Ius`arok]_ceiw[XlN@pbQ^MXtmpit>a^XxwofsIgXYgoCnWw:ueE]GnqiAywtEEc[roEG]QeiIW^whXIIEysBMbbGYU_vW_d`mslAIcmrVgDBLQA<L`yLl<nA<S:]mc]XXAUX]W_ivoeM[QYkpLfPuYiNh<Q=]rvuP@HWvMrhXvxicxOmxXwlfhR>\\SVgZGsLhrC?brwfHNZQpwyIh^Idy?buXqBVdDytYphTf\\<Q]hx\\fVwGn\\n?o:AbNc][iZEFvMWmYt^GV[qC;ob_aimyGNUR=cYd]Vf]UGkxqiBtkuPwDv=Tj?cAev?gRNqh:Qbosrpgbm?gPofact]SepCtmqhIsCMeYk?heYbRgBD;CRuiVay>=fflKgQuYyuieoUTKylppEvtArF<KIqm;@V[Im<DqFqq\\pUvmysdPdaxQLsq\\QV\\T_LrqHJQITx]KdiLetKF]TalME<Oi\\jH\\pw\\t=@sYuWa=KrEToUJJyJbXu=xKw\\vYUQdAw]EwXpUA=JwhwpxYhHsQlnJUyfyiLNgtYiOWt>NmFFy?`jIW`XicKyrs>jtxmuoss^pmYmEi\\dOyXNZ`ycxOxyGgH?woAtuisFpq>p`AAms?w;ymrytRWt<`yFqhsfwnYt<FikNiuhmnxpYq]B>];YhZNoY_sxnMoFdQw]]t][gyEFZOvtGuh;wYidekTJiiweROUR<=YH]bGIG^IbGEd]scvKtYWD?uw_[xLYCIYyKefdmv=YdIMtRwfvkTgOfkkrHobIaWxifyYEDyC<IcteccUVY;bI=DDiClYvQGxQsS>wsUyhvIDSWSGwSPEgFGWaawxuBf_XCWbmsWqKWmqD]efOUxpKgIUv`awHItksvWYui[F>=FbAVVyisIIoEucuWF_SM]wHKWZEVdcxZygXmX?gwgUr^mg>cC<GGByWsEvqiHx=Ej=CAixusYxaGWyx@iD=_V<kB:UG[eu^GbIwb`sx]qyYgyh]rqCg<mIa\\YyAsJYw]Xn]`pxDxutL`@uF]M<AjaaQwatMpqfLqQqquYy_epKTMwiMIIjU@nWmtwiRjqK_]xnyOLMJ<QWymLnIJbeQJASJaXfpXP\\tu]qeApIxqfQpEaTsmpVDmITmm@NYxsymytdX=XxyeuSylqEmkewFYpHQTaamumU:@Q>uvoipNEYGIlEPOG\\mbLN@hmAyUwqvqDJm]mMmSsdrsIpN<L:ENNqksTkr\\TbmpmtNr\\pe\\Su=vE=nQxJ<yK>@mCHxCdXO]rwAoIlX_yoVuqXINYdqx@t>dOLqrGXMctYUiYQuUwepyyYnxr;xJxpTwqN`pStqKb\\TGxXXIXU=WxAJxURXALJYr>hufXN^HwIaRU=nwIvRhVPImeYs[@X;MR@TymeSsDPnLWAIuWiOkmUFlkjip`esT@RWHx<DPiivx<rh<tOxYspQnpMbypEuQrEoe<XemY^lOcas?Lu?LrdAshHPeAU<MQwtONxJ<xoXyrpHQ:pYkps;tSJeJC=q]qsuLLoIpgmQKpN[XOYuTyqYJISADVq=namsVLKg]RTANamSVUMmtNjqM:Uikpj@Q\\mo`GairvtcwbvX[:_aoQp\\`yZVipxhOfZl>oFq^hVnxvgkq_EWaawsmovfNsbWiQflaO_GvsUqfqIwuw_x`ncXhPC>SC=gU]YvC_HZ;YiaYsquyUFLQD`mgoUuAosZKF:_hr[CJuSAyVh=e`mTIcgy?cBGSRWtroiPsY_UrIag`aVBYy\\ihAUgqmEqQi:qRQku?yhVQXEOixqriMyQAV`cXvIHAWtu]w<iXQCd@]Ryuh\\AB;]iyAvIIV\\erliIbiEsqhMuu`QspUTMMwMwevaDwWEyToeayHXyXiXpIux`KEhX[ItGUM=lLQ<rQhMSqlCDKpLuZ_f=iiqIqUFliwiuWstYi;yuPagehynxnlNvW^iPakvIrKybANtmXntGul?mPos\\OZ]WeJihZGd>QrChe?guhQqSHi;`s>hb<ItYXlCHkdfj;^kWasWXn[av=a\\ZFhR@]po\\AiqJqo>AvmIy>wx]yxIIa=n\\dwbB^^NqwOahKNhrqqr>xhaiR@\\]WgeYahaqM`ZYnoU_\\ppgWQc<pdNgmKypqwUIvQwfsMEJ?Xj?rn=seqCCkwr=TJiBgguWGxHoi=WJnISNeQjYMVTuAqnVeWCtsO\\k=UtKmYruvB]vR=roYMi=qKPxDTNYlVy@VQTM]mSYXJnXx]IKd@wq`nw]smIOhxQLpxiySVxkXmPlyOjmv`AuRAU\\IQ;ppWpmVtSNyqc=Kc\\pDETiyLJplnTxPlXCISayUgaqg`pfmupTqIuqqiuuPmttV?LZ?HZvO^f_j<HjGg\\L>mWNx;Fvy^_gnoYxxOYiWieyWs``ZxwpKxZrymZgcBVt:@`<AsAvvwGryQqUgiiWtiXj>hveveKw]yPxiHkr^empymNwPygyexSXsybNYFIgyLOr_qDnghSSe\\utyiuoQwnYhIyxWGsiIieqHoYduqiIuxw]vLyIaKdkqeEwIFEwZaXXWXBqCB]BYKXfwWbYxo;WJ?xXeRX_SL=R<?xy]d;swMiwTox>CWUEc?st:?bv=RZktQkUbeIj?XlWtGowAKBUqHRwV?Gy;?eECHYybWwhmysysexwyryrykCj@UNDp=dkWmUF<UU=MDaPseowdvNujCqK?lsfDsUMvaYmIIs@`rRyVLPRHQyH@LlUugxPAlL_]lAlvGTSApJOYpIqya=NHyQB=KUxXj]pk<mhAxwAWU=mTAlPMr;AubAotqywxxtQr>Lt[dQFyOneoNPXXHK:aKAusE\\vKuwAAT;YU]lUR`mryQFUjnEN@DyW=Y[iJNyu:\\S?]xWAT^hUx`YsdlGDigW`GGbUXx?qc=^cq>nSH^[xwpxiH^aMfx;qk;accfibatdhtIy^XXvM_hqvvsOyrvpKxcdYia`qtWimWyZpqqputWgZp_r^tmaqqwqU`jgA`uGdP>gknjjvZZa\\I>wsGp:WsIibUNfQ`jqvZFyv\\_s[_aGgsNPZw_[<^tFpapH\\cVpyggoEFitMwStCsQ;hKqHRWejSElsvNYIHAI=exI[CRUxsqxqmi=MuNsvg[drmIZoEKMrpoeyKw;;rnEC`aTR]efUxpigieYeCIlWu`iGXiSh]De_SdGIEMET_UvqR=UHaiWLoIgerVgDBLu=<LVyMXeqlLYMltsuLhUMX@NGyjH<WW<QIdQy\\RtMpSPO;tTYXqppj>=vX<c<a`Z>iYIrfYs`AnhoayYwI^ahYm]GvWik=FZ[hwrqsgvpdHnxqaDWhRInAPyxWj=Wb=oy`@uYyigqZDw[yakyxyux_@N^NG^Wnw=Qiwoy?wlrynVXbMoisybyNwL^pyhyUOqHNpminivic>nCHj?Prunusoltqxqpq[ylcQueqo]VtU?gR@fHQ[nPdmItP^Z>hZ\\__JOaZ>uJIrX?kn`fRyr]Gp;@xYG`poxj@j_Oomndmv^pvgZGjBV[YYmYax`fphgutHa;arKNuKIjro_fqnMav?FZYwuS`quqsUI_i^[^nhBOclXddWug`fLPru@uMVv]GZlpb`V\\H@lsVdhPtHFyxYygyb<H\\p?ts`aJnoZox:`aZYr\\XpT?[k^^B>ioIwhIw]Xbe?f<^rII^YY]A?l[Akw>p=xf[Qj?h]@HuqFdoOtMAjfObr^t?@]VNsAFvEhp^?pO`y\\ylx@oxnykxkuys>iabVmxamjvrSo`INj]IyNx<IryiX[uBsQrY;fX?eRqhxkb@AG;_GQUR`GTH_ghGB;?TRws?[BigxeyyDIFPgyXIf?eTdUYBkXfmr<_hroThcTd=SYoyVKE=mu;eRvkCR?Veiw<iWBGh>mSdIFceW:uE<?R?exk;XrYEHOXtAybAuxkCeyUNgs_kD[ix:OgtWguwrYIwAWVtoUpmeM=wuGUigwJqItUu\\mWv=y=yE`oh?Ee=qv=kwM=Xpss>chKCV<lwE\\u=Qu@=QrYniaWmqX@YPAekTxprmODyl`ms@HWtdxk\\LotWkYmXynYMylIPdYVQxYfpwxdRydp\\msD\\ZgYZM@eR@jZqyiysfaoKaivoxWw^M_ZcVxt^ymNrB>stXlPWeG?qZosSfn;FjPpr>hwsAyxasPiedicHFZ[W[Y^qYFmv?`XYh[YiNav^aeXxavxyApac@j`Y^f^ah>cBniAOev?vyyZLAnAO\\>>vuynAvb;Igeiqd>]GFhSqeEIsnhgxh]vy\\NaZEp[O^soGyXVyVIb^A_=@pPHuVgsHOaPV]fqxSq\\hVuH^`lYZ[?RORJAR=ST^SBaEhtStUID^Eb<uDDKRmQr\\GW^iBGYbxWcB_iVeW;eHPUiK[UVYHK[v^_SyWVQWTisCPSVJwChsSx?tGev=?b\\wwJKhssCDIfusCMgUqWhrWxZYFAQUA=y;qCqSB^mbCAFpMtmIy<gxFYcZqsywDH[HUsEoQCrEyOwT=KRK[wfIiv?GQkFgki==F>yWtAY:otAuHoKh;Sib?D^=Rx;DxWDsIr@cY@IY@[hAMIZYy=?gukv>mgAUBLsTymS=OheYVBaFdASmYfYcyb;TYkuWwRY?fYAtp;wIeU?SyUyuxqHJqs`Cy=mI^[WQ]UAgVngH@QDlQb@;rXot[kt?OR^MuyAYvIXy_wvaWnYTNuEMAygQsLmYQwYlYTocYFixjGDckGU;B\\oDOmHraFcIYwSxeyV<siMCW]GXLwxUqD_abpGRAshgkYISeI[XWYuWyvvQf@AGgmFrKbuCDQaGy]urUx`ai_qiaQH<khsUs[GYMosS]v_UEqOyRIceYTVygqIGXixoqxccBC=SjsVSUG];DBKIQUCl?t:=HG?XG;S@QYpwRRIYY[YWWba=uf[EZSG\\;u@siECY=uBTyCwiYhUBnSrBurUOIruyOaB]AsSYbm=e;ASPKBEKS:=hj?r`gUY=U[[bKWDrkDHuSBwItAeZ?V@?sf;EBSh@OsWqh=cS<eU@iU:YXTIfy=UZiW<UDDYDSwdq;C=CYscVnSx<wgDSCP_iU=EYqE[IWVAUMiUoCdA?FvGVaEeEYBmmBnWwkwrgGgSiSIsD:WhoSipIWpUwg=x]=yAatU?cOWt`yCvAyxABkOVMKEWagDwi=aD=uDlGvZUCMGdSACM[C`UytIu;[UI;DBqtGGxrqYISwUkDrAUmafuIS[ir>YV<WwLWxBMVZmgrcW=uteoXlmbWUDFyCBcwVqEvydCkcggCoyvTQtlAi\\syacsKKUiYVpCgKYyRWy`uwwsdPaxkWS\\Ws[iyUGCXgHKsTpcYA?Gd]emUu\\wERqePahmgSlsEvay<MyuQfoUwbmwoaXu]rAIWQkWU_W<KFUcHPQdHIUHDtvhsPqxchWXqPNynctvfMlpMLB\\JgYJptUDMXWhvc\\y`xYZ<w=avcdsmDXptjG<KuiMaYJ\\MJIqktmQu<Y=yu]yOxQv]\\LOQspHUlhY^]rq=QAHyNqyetKF`Yh\\OX=lH<rw=pJHtCtkUAWkYMcHSHIoeEKSYUjLxKmXILoTlMS@KjPRRipOhX;QSU=yGAXvhvl=P`QpGUs:qPGiRglWipOR`jsylnTPiltbar[LU=LTOEU]En@@LDqnslnjej=]WfMRLLt=akCuNpLymEtaDq`lO\\IXseTbuJvtLZUvR`o`EqFayJpjUQMqQvJ<rfxOZiVghYaqyP\\WepTYPTWakp`rhlQpEloylDIrAtMfykClLZaW`TkFPgwiZ=V`=gyJg^UFZD^kywb:sSaUlGIhWIlsVlgHw]EPqIhihFeSYSRmsTc_V=UsPktYoHYoGqUb;ieAqVF[XRiCiiyTgIcEbZYRwAigessWriAJ<]XL=VhpKHlKspQclRB@VPQKkYpKuMnyv<usUUXQdOxmu`xwMUurUWQXl_iJ<DrvQJ<LQ<]wh=yNIYnqovirmyoLIocAXtqS=<WTMrf\\M\\`PZDTsiR;LpTIJqxu=mKYAQvyK^Ev>YvHpnoPMltOFIobpYUQjHuu=HPaYX^YxsMUu<Rf=jgXupHt@ikXLVq\\S=LLZdUdEVplWP\\k:@xhepXlYQ<rHLMPTUP\\wgMRODmP<Q<UOUyVAew:LRJUpgAJtxNsuleuL>qL^il^uRkDWTdUj`Rf<WQ=RcQoctKlQjglPiAKl@LEQr=tQEhsm<NfEwHxytHUVUR>UP[]LXdUrpvWxSUqW`tkAduxELqqXsTLKuuwypCdmELRnXR>Lpn@Rw]xxxjJaS^lqLAyHiRmPsQUmimpchYFHQYaktQmUhL`]Jl\\j^\\VyQMq\\thmr]DNd<xv]lmHUauSjpvQqR>dua]sphuvhRwYUWyns@OMqP;@RJIx:=QVUT:xvjDq`iX>Er<YqRTm>uLwUxxqjFmp<quWixb=Rr`s`<X`DK:tQH\\RxPjIDNBQjE]RVMP<TLd<yl=ra@TQqrMXQ<mN@ULgHUe@NImqydo@\\nJhU^]J>lXHAxyAYTQvyITCqTe]tMMys=NJ\\rZuYw=vdhT;Xl\\hWEhL>dpSTssHNhQjRhoW=LwhjlHqsEo@hPZ<JvuUn]wqxxXmtlArvIJfqM^Xj?XLBlPtajmuKO\\OuHWvDNELmN<yS<mkLnUEOVQTsam@LtYxrTpNUIkOAMYTp_HuTHx:yt]dj^eWQYYx`jSpR>XYqunQiwYUtF=Kkar=AnUHwSXqFIVw=lN]kHTW=Aon`o[ESMyrOPScdXYtncxJahVM<NrYMjXxplmopYwyy^LT:Ilj`usQToiJyIyxtW<yPc]wZIYUQwxdsDIMZLJ;ur;tjqIRruS<mNJdJe=Y^mn@`Y>hmeILc\\y`iw\\yWwLMJXUx\\YPPy]YnPEoyeY\\<yxiUk<QaMkDtJkYrH@QL`sGIp@URELJLAjI\\mBINyLN<PoLquIHp`Ax@DRkpmbmM\\HJ[Ml<DquTmiqUV]J>xK?Ut]LQKIyY<PyMpflYfhYZ=rx=y:MQjToV@uhEQGLmSxTLay\\QtjXxfMT;hnJLplUPsDV\\MjWaJ>mwDPQ=@VxXukLS:Lp@UmD=V<Tl^\\NhUOdyUIDK;YS;hwfIlAHuqISHaUjIkjemu`sGpp[]UTAQMXRUYQ]HU^ukMqvlmo=`PvyOnAsk]MpUMgas[eVKdmiPRKPr`UWBdyv`WTaXraqgeXYPK>LjWYMh]LfxLe=sDlKxYSEpVtAj:Utk\\q]uRGHOR<kCIQEyUmtRppxYylPQM>]O>=vW<lK\\mGLreMQFIpZYt;LT<\\kImS>mJhDJ;EvoTKT`l;iQPHULdNjIJCaO\\tMJ@pjYqZejtqwg]LJpK[Ysnur]IynHjheRxEqBuRkqvNxjoTSJ<Vjmp;pY`<J=Dt;axjmmEDtHHTutmSaJaMoRESiMr=itb`uOQSUak]pTEDVcmXSXtnHNSywA]xI]tmUwcXXwLMJLK\\DTh@lshLFqojmqAHQhGg[ai`or?v\\V?bqAmawsnP\\uXrvhgowmD^Z<ps=VmcFw:p`=NmKV\\r>ni^_I@pB_dp>b]OtFw]JNipqmEO\\axrtHr^_ZiagHVqUYty`rqIw:Ib;@xCFtIwbqFxDqxgqy;PjjI_;_`gPmb?nC^i:on`Wsdxhk@\\Jn[yXyMybXFoS?wMPZcGpuodHHamOnKBemhJwtBMe`]dAwFVKF_H[whqS?q<HsC^jk^]pipJhwRf^f`dfvnDyjsFeANt^OhCYsWa^AwdNh`h^Zm_l=wgpVmcHxoO:kgrOFKYx[Od_aY<Qy\\ccZ=e`gC]awQUthqdqSTbmtMQVk;b:wcm_HjmYIIUaaiUyTMWwm]boOFrocJOhfAgXQWLKhu_sUqTqIg:=FbaSRoyX;FYSUpgt@GcGOt@[f;keOYuuMeweIhobY_bHwXRIBc;c@oi;_D`YHBEfpeI<Khnucw;wBcBrqb:UC<SSYkXJuhPYrSufRUU<gX:CVIYWBgTFoB>otMOxfQbnYy<AW:]E[_C>[FZAGMuBZuxFWXs;GfugQpW]<RALpjptb\\svpvUUJ\\xmy@Jj@NEhMNUTBHvMhugMjhUnBMThUPDImIMPt`w=Ytndw;iq_\\St@M[AQFyY[PTcemtIPCHsLTyB=uVLY=QuJDnchWxyQ[=rExWp`PWlyvAlxESTqjpMVOEkfYvnTq_aVLpRTqsbEJqdo@\\RaDNRxORxPiHUp`kN]vrTj\\Isatm^hkLdUe@jt\\Y^YRQPoUpWtFlP@omyfLW\\PWsR_w^HiUawxfktw[Nqng@_;@bjXiu>adgn:>ZVFjC_b;_dZfk@^vdXqX^cWxyqo]TagooiA`gTfn;FigqmoPq<@pJ^qap]sPjxQv^^hrYayxx\\Ycdo[\\^vG@irI^<@bV@i<I[daymvy>@cBixqr^WHIoxOasr?ERAh^UTf]fycyFwXJyxLGS:MVn;FJAfFesPwD^wcaie:?T^qEaYHlkie;hjEfkIr>shuccD?tZ_yQaBFEFRkyCuiHmYSqgfeUu[W[ubk_Hmsg^Gy?me=Kvf[rVcxdCCsAbjMc?seC;DEkHrSw>uWqYWiWihmGlAeQoilscT_fV=smQfpMugmV:eUsaY:=DmGCIqhvcUU[xVUTdQWmiSKYr=sRqWRh[t[wDHMi`wCNwGrSEHoI>]IGsiMKISQiCMrS=Uy_RowHAEceGgmmFWorIkifKSE_BO_HySUSaxn_wRiSQiexOX=quP?s>_cIEIy_wlWsBIxYuxwGbsoDnkFcURjqd=qHs[iMStGyx_UuHYulEi[aI]Qyn[WvghlUt:=FJ[GfCu:od;CvoeSZiuFsrhUtceiX?WREVHSwRUT=aHZ;uvsbC]tOyvhgCVyWumIvyIR_WZ;RB;C?kHSCvI=cocVruvBOy[iYy;FmYGJ;cEwuqqIDUCmKgIuUkeFwww[McoEw_swMuEr_xLMIBsdGmbYcYnyv^SC]yx]ycs[enKxfMg?WbTexoOfNmhuQibQePWuheiSotl[vIWTSYYv?W]UiWCB[cOaPVb=y:UVVpXTPUsPV?]nZIk?uSYIX]hM]=m[EYppvpnhBp]>heDWu<nf@Hl^hZKIq@Vioqf`i\\a^gcixEP\\vA\\oQ_CGiVYl?X`dqjUnqdylCv^NVrcwuvfhXpsM@_Q_uMykeAawYqKPacEcARSCFrIw;kB`abRsxFodtMrGOv_ITbovVKiKQeCgVUSikmFE]w=abQKcH;IvSw[AgAKR\\IbUgwmcHMeFu?TwAVAMISYI<=SpGWo;wnWXEUISQuj[Ggse=KvLCgVeSYsEs]vKGfw]ckUdpwcMAya?XTUHFggMYVHYCv_SmIfBWxw[Ig?CBcITITqIhZASqMYBWw^ysi\\RPExf]quLulmu`QUYynpHpSasGUJ;@RjtqjhRvPY]Qk;AriAN:lPG<sRarNDW=<vnmW@EbP@kR`^<FivoxWw^MYqEB>qTHaiamSwEySsW`wiqyWxmGn[bjIUemUiyU=gRFKfXExS;Ft;RxkIueKNAj`HM;eMiMyAYmOypyhxGdxIupYYnxTpxQMDDvEAVyxkIaUxINoEvtXk<lY\\<xYerTDrcMrVhwIEQ_aLtxM><KYlUQAnSujYXVHepKDJSdMWdTi\\qrQVbIo;IV<LLVPWn]SHHqqYmHqvBupg=YaPULYUnajLaod@Q<YUxTwt<pHalePWdMWd<XX<KxqQuhVEDYxyTuQYNqokISLambArcHwAeMw]odplu]T[XYHqYJ\\wMarlivEite@O=ULmQMjQwmLwF]MmtqQmpiAxReyJPmSpmppQMXtJTPqlKV\\vlaTDeRt<l@yOIAsbdjRXtGqRJyWbtnsMwvLxilLEPoyxtGtRDHKHpXpDjt@NImNEDSLirOIOQhM^Tqp=oyasm\\T@]qkmpn=PrMkoLw@DVEaxQ@R>=nXquXLLPqUSqQVUVeEkl\\wnMwYEXtYnJ=OrYq\\yOsTVTaM]XUqpWOuOw@xvdUIMMcAW^urpUwoyuwtxfdkImMAlqZevKTtQuXthxNiy[@xlARw=w^Gm;vng^utVw^Y]KGpJxg`PuAOuYNusAxTYycynX`xqOwkw[N?qbY\\UYkBfw<Wxh?fZ>`xpjq_wnpnOYuJxvcNtTQuUFd;OcHoqRIj;YpGqa`Iht`rcy`WVq`n^cfqSyrvnp`^mYigIYvt_xpwvoHpp^]TphSX`pppuffRqxuy\\B?`@YxZOy@pxTpnsFdnI[lalMqy]Vq`>w^Aiw^eiNs>pq;Yh=qv=paHHblnb@wre?[=ApQo^Nh\\iy]yqyWhpiYstwvgOln>^@gnSImJIiEPhaqxVOwqIlgxcgWxpHlxf]dyw_vrJOnApsvh[UXpUwetGdgQ[SwgfXgTqoWOcZ>\\Fy_Z^my_xRhndIocHZo_bk_d_whx@opteMxyuycIhxiHvWduyXGSDC=ryIytyWxyeuGbkOhEiSIGYyQsZIrnYESmsbEy\\ywDqgKehCIdM_W:MEbEGXwuruedsEr]eYmdGUeX]Tu[equCDAclOwIyy_yGX?xiLxxHYaQmUpj^ImauPsIyfTUTxKfMO<Qn;tv:@rdujqdjkqy=QY<PXyTpsylPQP<QOBDWJuSweK:hrJtuuhvqtqPUUFXyZhu:yJrErKHRjIKEyyEpSX<YiTnV=OM\\VQyrYTYfdTYPRxyrc`w=TJ=dsa<NJljZ<V[inIDnAaL>hXutxVppelV@mq[@w<lsY=KDpn]dN>HjSQkMElx@njLrapuP\\qW\\UwlkvDK[YMqLomhM@qlCivOuWYewhhMtlKELQDAsa@V]Er@QvGhJEuOAlSQumddlWLXK=TwPok`qHmQcaNqLp:uWYdw\\mLgTmm`vj\\ThPTXeUsQjMxSgqobAt:uM_YjpXsPUs;DNNdv;Yp[APWYmi<yuXjrpTBIMaPKipUguMrlkJhlgQOrix;uuPLXwdP`aVbTOBDNaXwp]JJaYPqUalSQlTaloq\\tw]n;IsGaJ>TQviq^QvEhupHRwhqk<roURShXleju?qpWbg?uOXnkit]GkkikeAc=XhM>m:@fvVw<V_MxbTvvHYpFyu\\v^MhvSXspQeHOrtwp`AoF_jTIwMIwSH]dv`Hw_oVrKW`[>sufgRYuQYo@^kb`eU_]\\ii<X`[InGNvxFbuw[nqZUY]i^\\NIZkoqEGqLgsr@nknvB>is_xap`iXlnHfVOeNIZ]anUg\\Nah>w]aYx;w^hqoVaodi_^YgtA]wIjvQqVPaAsBiX?ydGiSUiwNGTcwfoSUT]siwXWgYpceFSeMqSZqG\\AfEQVk[XxmbGyBFgTvwtJuRtcgVCYOcuqmbg_SHuFdow\\uhPqbpescsh?MesUcnqhpURTkgiSGhKraiWKOwLytMGeQoI?ayByXqAy]iCmqHhms\\gGveWRAtLOEM[ilwVtkhIquuufLQeC[B<or;qt>YWFYw@iHwyye]WsabVUHiIXwYEducj;u]ihi=Temxn]vGYRQib`iEoayTOHY_XR]XwGdpAi]QtqwdqyD;oRLquM_gykCh=gyCtx?uROb<OBiAVUwiD=E\\eXf;tv=bv]GYYWiKUF[SlID?Sx?yIkcwhQywKicUVZQI>kgCIUQaHEsxbyxk]ermUbucrKvB?rw?V??DbAVhIVucsKsi=?HZGdVUyrQhTOekeBk;dQce>cdvsVpMFvCCmQeRoe[Cr=asEScVwWHEdj[fegsQ=bQ;S>IW@EyJqTF;CG=Ypac];s\\aW<GuSIEokIocSKcCXkycyS=cv;Ue<=H:?RdwwtOi^Kg\\ihEaGOiCceW;KBFAewivJSEj]fUceV_x\\;R?uCkKHWEXDYSK?Fn;XsyRAyFWQXssEvuHogB^;sVCX^CCJESFAVZMt[mcEqVCCF[MgHIckICTIw?yVaAwROukWWNOvuUfXYi`kfNicq[uR;SIKRWMx\\wV[gGs]SdiYIyBrmSKHND]x[xwOdLr\\uMTQtAL:Un\\aYIAW`ARtayETwYmW=AqBYVfUYDpY;yxjpQXDMUtnkXkIlrGpyU=nIXs_tQiTumxVL=y?\\TdlW>dQ^ipdetVamZMXItQoLmMMUUQV=<J^xuVxvVqNmHL`=TbXNGisRPPciY:iX^PPTTxPTxXlvj]TptWXQvwALQ<u]`TddjyXwWppIPO]LsM`nIEtHesdYtLxpK\\TmpORmvfTWO`jN=K>Ljshs]`MS`W;Uu;dkDly\\YTIIWaQJuUy<XM:XN:XOfausXPkxsc]o>qrfpnS<jtyMdmxahKu<VbePZ\\qcQRypqZARBTV[yLElQPlYW=VbQxpuLgMJBlYyXvHYQcEr@@L;Lr:hJlUx`is;tjL=Kq=tl\\vvdWH]UXULHQRGTl@<uUInxlVX=L>DNrlTvtMOPTB=wR]UaAX<IVK]UR=M?Emj=TGpkiQQIiwLeTTiSY@mR@jN\\S<eOVpm<dvkqLaeQtPLpUN[MM@Dn>dX;mMZlVYuy@yPdMR\\Tnf`Lv<rK\\OR=vd@UvYMJxLTaJY<K?mptDwoqNgDUhtSgiJoPN=lxH`VhLvBDwJltwEjWIN>QYWAOl]jm<ykXjD`q\\ajaanbxr`pu`HJ;Am@qWgtYjyYv=LYYMH]sBpn^xpo\\mGAtP<oITuKyLtPX>=toYTLEtq=Mq=Wa]k?]YjiMdLWwulSTv?prImPV=smHLm@PldVgYsSQT^UY]xUJuM^HpoIP`YN@qOx<MiDXw]m^`SC=kNTJdXVWtXwuxu`PPAWHEYgUklewVTmbhmaqsrDruUnNesRUnG@xvXRWDpK=JlISwITGDNiqrvEpK<YXxpBUklHnV@YadYeqnU`o[`sktNsmu`QXsTWuExVhRTIKiUL]\\lbDYmqLwIVUxpRAW<eyZmX?xS]hYTHoPyQQXrnxM<DjZExZal=ulMyvIySt\\syqq^xpbdKMaV:XOv`xlYsXYqpPLE]R\\eSj@VaTK`UlN<Y;YuPDQW=omtXcqTW\\ytLL\\=WAytB@L=YKBdOr]LK<oN<x=DJJLvfesUxkIdtHARb=rhHNAEu<qO`pOB=pSiJPlSR@j[Im?@La\\SHYK`YJ:hOi\\wZURZ]wglnFHUgEUnIqxAL@EpRuOYyVyLUvHUBarmHnSHVuuQphLNYuAXs]`M;yrnDVupxdysKUY<aj;uqQuY@lyIdo=\\wGtKxey[]pEMu_EWX\\nuPwdxqlyyI@WIDSNYYbytkHpkLQyERQlmKdy;hWWDJB\\u`INe=KX=jWMt\\XlAxq]]KlTWkdm<ixLuXKpJweN_qWsxOJlR>htodjp]lods[@Xjmy<PkPLJdAt:uYGuu=anCuwQ\\R;AtUyrVTkriK]ijlAkCdVDDJtDkwlnXqjC=vLTmfxtAYTkhYfmt[\\oIXnnXPtqqD=wR=RxmSTqOIIoJamMMyVtKUTYxaQIIqxLW`ay?aY:=wVeYx\\wyPx:MNdppWMYNHoEQRZuN`eUGUjxay^XxPLUAlmq@tnMmKTYKpvDXOh`KNHLiYv^dyU`XXqPTyntAuFXmqvhDFasNqJ?^J>[`Y]JHrY?hrVZfNGawRaba?fBEFP?Hgsc<gb?iRYIe@WTZGwyIHQ?uGSRCYcFGS>=hM[IocI?UitOy<oB[ABE[hgkdkkXNGhesD_[C\\Et?myWmeyGBFUETCsEKRC[BU[dYOwYKh=wb?sD<KDaGGS]yH]x?EEWWI`ACyAUjIR\\aXa]YwKR<oCV;UA_IR=tt[vJWR<kY^[YrOeH[yhmI=Cy]QytyixaEgQdP[EleTlMkfYKwtOJhqOELoMvkDTf\\XcHOA=O=@V@DNVusiLogHlP\\vr`v_IlB`S:Um@QRF\\p`mKclklTO>@nRlVLIl^akI`P^YvWtwvTknLJ=pR:uJnX\\CYrkvywxipyf;piCVgYnw=?[^gyHwesQmDV[kproQc[Pd]^u?htu>t@H\\cghlPrw_kppcUyrjYnKOeYv^[>mCOs``mpGbsoHWvNCS^CSuOwmQcocTBoBO=T_epSTwkUx;\\J^lmjqswxyJ<V:Tx<\\nZHMmqL]INxmsd@tqELyxoTdmqHWtatVTmkUXPauvHpwXS?LJ;@RJQMrPk>tviPmmDNqPsJUxK=Mf\\JMpuhAphYO;<rpHME`kimq<LL\\ImUEnLLy<Yr>`rnDmZekDhsVEJHAvR=xf=r<aK;MJudXJlNtaPGhjgtOV<LMtlQtwryxeMww\\wZirS`p>en[iRYXnWDrFhWwQyyPKRpQ>hlALjE\\Sq@TX\\tCtuKTPt\\wX]uAxu[pm>DmdHMj<ND]VLYvpiJ>DkjpPC`oiHTQ]R<Mro\\M`XjluJBHWBeuJpL>EL<lPIhLXtOKpXZQuRTT[LrPdUAHmRHlITLLdyMuOJuKnhuoyUwHtgXk=mQZatFTmrUT:yro<QRQREujJPsKeYIxuJyjwpUnEoh`YElX>hkp\\S<=yJyrbyonhYFHXKytMYmD`MSxwH@pstjpHjMxQ:DNwLoWEqV=wi\\smMNq]McTsMQy`DmI\\RN`lW@KdqnT`UA<lV\\y@pQGTWm`Nmql;LtXdT@MOAMRM\\qtuvgMmxdKFxPi]sO`T`dLsAKUaUJ<K>LJ;Ux_YJfhUmMugUyupPMeksUT]\\jS]mSdkguNRaP=Qj:xKu<VP<no=VAUWb=WJeLuPRpepvlJ?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Tips and Techniques: Linear ODEs and the Method of Undetermined CoefficientsRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoft
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>The constant-coefficient, linear ordinary differential equation is standard fare for a first course in differential equations. The solution of the nonhomogeneous equation is a sum of the general solution of the homogeneous equation, plus a particular solution of the nonhomogeneous equation.The particular solution is generally found by either variation of parameters, or undetermined coefficients. Variation of parameters is applicable in all cases, but the integrations it necessitates can be formidable, even for a computer algebra system. On the other hand, the method of undetermined coefficients gets by with much simpler calculations, but is valid for a restricted class of problems.The manipulations in the method of undetermined coefficients begin with a trial solution, a "guess" at the "form" of a particular solution, which is just a linear combination of a set of functions determined almost entirely by the inhomogeneous term. Although the ensuing calculations are merely a substitution of this trial solution back into the differential equation, this method is difficult for students to use because the rules for determining the trial solution are not transparent.These rules can be reduced to an algorithm, but without some insight into the basis for the algorithm, students seem to find the whole topic opaque. Through the many years of teaching courses in differential equations, I have tried a variety of devices to make it easier for students to comprehend the connection between the nonhomogeneous term and the initial guess for the method.The most recent versions of my approach began with the caveat that the method applied only if the nonhomogeneous term was itself a "fragment" of a general solution of some constant-coefficient homogeneous linear ODE. The full solution that contained this fragment was generated by a set of characteristic roots. The "trick" for the students was to determine what those hidden (latent) characteristic roots might have been.For a long time I thought that drilling students in the "forward" process of writing a general solution from a set of characteristic roots would lead to the essential insight. Alas, I discovered that the typical student still had difficulty thinking "backwards," from the nonhomogeneous term to the characteristic (or latent) roots that generated it.For an equally long time afterwards, I provided students with a table of examples where sample right-hand sides, the latent roots, and the consequent trial solutions are listed. Table 5.10.2, reproduced from the electronic version of my Advanced Engineering Mathematics text (Maplesoft, 2005), which is Table 5.9 in the 2001 Addison Wesley edition, is an evolution of the classroom materials I used with my students.
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(8) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSFGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRkovJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RidGK0YyRis=LChJImFHNiIiIiJJI2J0R0YkRiUqJClJI2N0R0YkIiIjRiVGJQ== (9) 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(10) 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 5.10.2 Examples of nonhomogeneous terms and the form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInlGJ0YvRjItRiw2JVEicEYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= they require
Unfortunately, this approach explains neither why the method of undetermined coefficients works for nonhomogeneous terms of the types listed in Table 5.10.2, and nor why a linear combination of the terms generated by the latent roots should be a trial solution.On the other hand, I learned the method of undetermined coefficients from an early edition of Rainville's Elementary Differential Equations (The Macmillan Company, 1958). Rainville's treatment began with a symbolic (theoretical) discussion of the annihilating operator that would convert the left-hand side of the equation to a higher-order equation, and reduce the right-hand side to zero. The general solution of the new higher-order ODE was then a sum of the homogeneous solution of the original equation plus the "guess" for the particular solution needed in the method of undetermined coefficients.For several years I looked for a Maple approach along the lines of my decades-old Rainville insight. I wanted tools for automatically generating and applying the annihilating operator, thereby illustrating the theory that made the method transparent for me when I first learned it.The calculations in this article demonstrate how the six software tools listed in Table 1 make it possible to teach the method of undetermined coefficients the way I learned it. The code that creates these tools is hidden in the collapsed document block in the following Initializations section. The block can be opened by placing the cursor in the blue output ("Maple initializations complete") and using the View menu to select Expand Document Block.
CommandCommentAnnihilatingOperatorCreate an annihilating operatordisplayopDisplay an annihilator as a differential operatorApplyOperatorApply an annihilator to either side of a DEExtractCharacteristicRootsList the characteristic roots for a homogeneous DEExtractLatentRootsList the latent roots for a nonhomogeneous termFromLargeSolutionDeleteSmallExtract trial solution from homogeneous solution of annihilated DETable 1 Six Maple commands for teaching undetermined coefficients
<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>restart;interface(typesetting=extended):
Typesetting:-Settings(prime=t):
Typesetting:-Suppress(y(t)):
with(DEtools):displayop := proc(r)
local b,x;
interface(typesetting=standard):
if r::'`+`' then map(displayop, r);
elif r::'`*`' then op(1,r)*displayop(op(2,r));
elif r::'`^`' then
b := op(1,r);
x := op(2,r);
subs("QQQQ"=convert(x, 'string'), Typesetting:-Typeset(d^QQQQ/`dt`^QQQQ));
elif r::identical(Dt) then Typesetting:-Typeset(d/dt);
else
r;
end if
end proc:AnnihilatingOperator := proc(Q)
local F,a,FF,res,k,qq,qq1,qq2,R;
FF := 0:
k := 1:
while FF=0 do
F := add(a[n]*Dt^n,n=0..k);
qq := DEtools[diffop2de](F,y(t),[Dt,t]);
qq1 := eval(qq,y(t)=Q);
qq2 := solve(identity(qq1,t),{seq(a[n],n=0..k)});
if qq2=NULL then return "An annihilator cannot be found";end if;
FF := eval(F,qq2);
k := k+1;
if k>5 then return "An annihilator cannot be found";end if;
end do:
res := remove(has,indets(FF),Dt)[1];
R := sort(eval(FF,res=1));
if lcoeff(R)<0 then R:=-R; end if;
R;
end proc:ApplyOperator := proc(L,Y)
local DE,DE1;
interface(typesetting=extended):
DE := DEtools[diffop2de](L,y(t),[Dt,t]);
eval(DE,y(t)=Y);
end proc:FromLargeSolutionDeleteSmall := proc(B,A)
local NA,NB,C,LA,LB,LA1,LA2,L;
NA := nops(A);
NB := nops(B);
C := {seq(_C||k=1,k=1..NB)};
LA := [op(A)];
LB := [op(B)];
LA1 := eval(LA,C);
LA2 := eval(LB,C);
L := [op(convert(LA2,set) minus convert(LA1,set))];
add(c[k]*L[k],k=1..NB-NA);
end proc:ExtractLatentRoots := proc(X)
local R,DR;
R := rhs(X);
DR := AnnihilatingOperator(R);
solve(DR,Dt);
end proc:ExtractCharacteristicRoots := proc(X)
local L;
L := DEtools[de2diffop](lhs(X),y(t),[Dt,t]);
solve(L,Dt);
end proc:printf("Maple initializations completed");
<Text-field style="Heading 1" layout="Heading 1">Undetermined Coefficients</Text-field>To illustrate what a student experiences when faced with a differential equation whose particular solution is to be obtained by the method of undetermined coefficients, consider the differential equationPkkjREVHNiIvLCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYlLUkieUdGJDYjSSJ0R0YkRi5GLiIiIi1GKDYkRitGLiIiI0YrIiM1KiYtSSRleHBHRiQ2IywkRi4hIiJGLy1JJHNpbkdGJDYjLCRGLiIiJEYvThe first step would be the determination of the latent roots 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 by inspection of the nonhomogeneous term 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. The next step would be the determination of the characteristic roots for the homogeneous differential equationHDE := lhs(DE) = 0;In Maple, this can be done by obtaining the characteristic equation arising from the "exponential guess"EG := exp(lambda*t);Substitution leads toq := collect(eval(HDE, y(t)=EG),exp);Use of the zero principle for real or complex numbers then leads to the characteristic equationCE := simplify(q/EG);Of course, the experienced student could write this equation by inspection. In any event, the solution of the characteristic equation givessolve(CE, lambda);for the characteristic roots. Thus, the latent roots repeat the characteristic roots, so the appropriate trial solution for undetermined coefficients would beYT := t*exp(-t)*(a*cos(3*t) + b*sin(3*t));Determination of the coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== follows from direct substitution into the differential equation, a step resulting in the equationq1 := eval(DE, y(t)=YT);But this "equation" must be an identity in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, in which case the Maple commandq2 := solve(identity(q1,t), {a,b});yields the appropriate values of the unknown coefficients. Of course, working by hand without any technology requires matching coefficients of 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 and 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 in q3 := simplify(q1);Maple finds this easier to do if the factor 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 were not present on either side. Hence, we are led toq4 := simplify(q3/exp(-t));and then to the equationseq1 := map(coeff, q4, cos(3*t));
eq2 := map(coeff, q4, sin(3*t));It should be clear that the solution of these two equations also yields the particular solutiony[p] = eval(YT, q2);
<Text-field style="Heading 1" layout="Heading 1">Using the Tools</Text-field>To illustrate how the set of tools listed in Table 1 work, again consider the differential equationDE;The annihilator of the right side of the differential equation is obtained with the AnnihilatingOperator command, but displayed as an operator with the displayop command. Together, the result isRK := AnnihilatingOperator(rhs(DE)):
displayop(RK);To verify that this operator really annihilates the right side of the differential equation, we need to apply the operator to that right side. This is done with the ApplyOperator command, and results inApplyOperator(RK,rhs(DE));The annihilating operator must now be applied to the complete differential equation. The right side will be annihilated, and the left side will become a linear, constant-coefficient equation of higher order. Again using the ApplyOperator command, we getnewde := ApplyOperator(RK,DE);This new differential equation dictates the appropriate guess at the form of the particular solution. On way to trace the connection is through the characteristic roots, found by the ExtractCharacteristicRoots command.ExtractCharacteristicRoots(newde);The characteristic roots for the homogeneous equation determined by the original differential equation areExtractCharacteristicRoots(HDE=0);Thus, the original homogeneous equation has characteristic roots 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, but the new equation has characteristic roots 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, and again 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 The second pair of complex roots are the "latent roots," the roots hiding in the nonhomogeneous term of the original differential equation. These are the roots that students find so hard to extract by simply inspecting the nonhomogeneous term.The latent roots can be found with the ExtractLatentRoots command, as we see from ExtractLatentRoots(DE);However, students would still not understand the connection between these roots and the characteristic roots, a connection that becomes clear only in the light of the annihilating operator. Hence, we obtain the correct form of the particular solution by comparing the homogeneous solutions of the original and modified differential equations. These solutions areYh := rhs(dsolve(lhs(DE)=0,y(t)));
YH := rhs(dsolve(newde, y(t)));The correct guess for the form of the particular solution is obtained by extracting the homogeneous solution of the original differential equation from the general solution of the new differential equation. This separation can be executed by using the command FromLargeSolutionDeleteSmall, and results inYp = collect(FromLargeSolutionDeleteSmall(YH,Yh), [exp,t]);which is the appropriate trial solution for the method of undetermined coefficients.
<Text-field style="Heading 1" layout="Heading 1">Comments on the New Tools</Text-field>Maple's DEtools package contains the commands diffop2de, and de2diffop. The first converts a "differential operator" into a differential equation; the second, a differential equation to a "differential operator." The differential operator is not displayed as an operator, but is represented as a polynomial in some symbol such as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjRHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=. Thus, given the differential equationDE;the differential operator embodied by the left-hand side is given in Maple byde2diffop(lhs(DE),y(t),[Dt,t]);On the other hand, the differential equation generated by applying the "differential operator"PkkiUUc2IiwoKiRJJ2xhbWJkYUdGJCIiIyIiIkYnRigiIzVGKQ==to 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 isdiffop2de(Q,y(t),[lambda,t]);The operator that annihilates the nonhomogeneous term 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 is generated by brute-force application of the diffop2de command. Linear "operators" of higher and higher order are converted to differential equations in 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, and at each stage 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 is replaced by 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, the resulting expression being tested for identical vanishing as an identity in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. (This step uses Maple's solve/identity command, and provides the coefficients in the annihilating differential operator.)The displayop command, written for this application by Dr. Laurent Bernardin of Maplesoft, then displays the annihilator as a differential operator in the symbol LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkmbWZyYWNHRiQ2KC1GLDYlUSJkRidGL0YyLUYsNiVRI2R0RidGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZDLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRis=. This command assumes the independent variable is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, an assumption that is actually first made in the AnnihilatingOperator command. Given the difficulty in programmatically distinguishing between the nonhomogeneous terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRImFGJ0YvRjItSSNtb0dGJDYwUScmcGx1cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGJy8lJ3JzcGFjZUdGUi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLDYlUSJiRidGL0YyLUY5NjBRMSZJbnZpc2libGVUaW1lcztGJ0Y8Rj5GQUZDRkVGR0ZJRksvRk5GLi9GUVEkMGVtRicvRlRGXW9GVUZYLUYsNiVRInRGJ0YvRjI= and 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, we arbitrarily opted to make the independent variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, always. (The alternative would require explicit declaration of the independent variable as part of argument lists.)The ApplyOperator command is just a front-end for the diffop2de command, hiding the difference between the displayed form of an operator and the "polynomial" form required by Maple.The ExtractCharacteristicRoots command uses the de2diffop command to generate a polynomial representation of the homogeneous differential equation. Of course, this is just the characteristic equation, so the characteristic roots are immediate.The ExtractLatentRoots command applies the AnnihilatingOperator command to the nonhomogeneous term, resulting in a polynomial representation of the annihilator. The zeros of this polynomial are the latent roots.Finally, to extract Yh;the homogeneous solution of the original DE, fromYH;the homogeneous solution of the modified DE, we must do more than simply subtract because there is no guarantee that the same arbitrary constants will multiply like terms in the two solutions. To deal with this difficulty, the FromLargeSolutionDeleteSmall command sets all the coefficients to 1, converts each solution to a set of basis functions, and performs a set difference. A linear combination of the resulting functions is then the trial solution.A final word to the reader: The six Maple commands discussed in this article are not "bullet-proof." Any of these commands could be easily broken by applying them to inappropriate arguments. As even a perfunctory inspection would show, no attempt at type-checking was made, so "garbage-in, garbage-out" is definitely a warning to be heeded. We leave it to the reader to improve on our coding - it was our purpose merely to demonstrate that Maple already had the rudimentary tools upon which a superior pedagogy could be built.Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 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