2nd_order_ODE

2nd_order_ODE - 6HFRQG 2UGHU /LQHDU 'LIIHUHQWLDO (TXDWLRQV...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview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  $ YHU\ VLPSOH LQVWDQFH RI VXFK W\SH RI HTXDWLRQV LV \ í\  7KH HTXDWLRQ¶V VROXWLRQ LV DQ\ IXQFWLRQ VDWLVI\LQJ WKH HTXDOLW\ \ \ 2EYLRXVO\ \ H W LV D VROXWLRQ DQG VR LV DQ\ FRQVWDQW PXOWLSOH RI LW & H W 1RW DV REYLRXV EXW VWLOO HDV\ WR VHH LV WKDW \ H íW LV DQRWKHU VROXWLRQ DQG VR LV DQ\ IXQFWLRQ RI WKH IRUP & H íW  ,W FDQ EH HDVLO\ YHULILHG WKDW DQ\ IXQFWLRQ RI WKH IRUP \ & H W  & H í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í  ([DPSOH )LQG WKH JHQHUDO VROXWLRQ RI \ í \  7KHUH LV QR QHHG WR ³JXHVV´ DQ DQVZHU KHUH :H DFWXDOO\ NQRZ D ZD\ WR VROYH WKH HTXDWLRQ DOUHDG\ 2EVHUYH WKDW LI ZH OHW X \  WKHQ X \  6XEVWLWXWH WKHP LQWR WKH HTXDWLRQ DQG ZH JHW D QHZ HTXDWLRQ X í  X  7KLV LV D ILUVW RUGHU OLQHDU HTXDWLRQ ZLWK S W 7KH LQWHJUDWLQJ IDFWRU LV X W  P W ³ P W J W GW ³ X W GW  HíW H W ³  GW 7KH DFWXDO VROXWLRQ \ LV JLYHQ E\ WKH UHODWLRQ X E\ LQWHJUDWLRQ \ W í DQG J W W ³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cc  7KH VWDQGDUG IRUP LV 6XEVWLWXWH X c   X W ,QWHJUDWLQJ IDFWRU LV X W W W\  \ ([DPSOH  W ³ W GW   \c W W S W W   J W W W W  · § W  ¨ W   & ¸ ¹ ©  W  & W   )LQDOO\ \ W ³ X W GW   &  W  W  &    W  & W    &   ,Q JHQHUDO JLYHQ D VHFRQG RUGHU OLQHDU HTXDWLRQ ZLWK WKH \WHUP PLVVLQJ \  S W \ J W  ZH FDQ VROYH LW E\ WKH VXEVWLWXWLRQV X \ DQG X \ WR FKDQJH WKH HTXDWLRQ WR D ILUVW RUGHU OLQHDU HTXDWLRQ 8VH WKH LQWHJUDWLQJ IDFWRU PHWKRG WR VROYH IRU X DQG WKHQ LQWHJUDWH X WR ILQG \ 7KDW LV  6XEVWLWXWH  X  S W X J W  ,QWHJUDWLQJ IDFWRU P W H ³  6ROYH IRU X X W  P W  ,QWHJUDWH \ W œ X W GW S W GW ³ P W J W GW  & 7KH &KDUDFWHULVWLF 3RO\QRPLDO %DFN WR WKH VXEMHFW RI WKH VHFRQG RUGHU OLQHDU KRPRJHQHRXV HTXDWLRQV ZLWK FRQVWDQW FRHIILFLHQWV QRWH WKDW LW LV QRW LQ WKH VWDQGDUG IRUP EHORZ  D\  E\  F\  D   :H KDYH VHHQ D IHZ H[DPSOHV RI VXFK DQ HTXDWLRQ ,Q DOO FDVHV WKH VROXWLRQV FRQVLVW RI H[SRQHQWLDO IXQFWLRQV RU WHUPV WKDW FRXOG EH UHZULWWHQ LQWR H[SRQHQWLDO IXQFWLRQV‚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‚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í  DF !  7KHUH DUH WZR GLVWLQFW UHDO URRWV U U  :KHQ E í  DF   7KHUH DUH WZR FRPSOH[ FRQMXJDWH URRWV U r L  :KHQ E í  DF  7KHUH LV RQH UHSHDWHG UHDO URRW U 1RWH 7KHUH LV QR QHHG WR SXW WKH HTXDWLRQ LQ LWV VWDQGDUG IRUP ZKHQ VROYLQJ LW XVLQJ WKH FKDUDFWHULVWLF HTXDWLRQ PHWKRG 7KH URRWV RI WKH FKDUDFWHULVWLF HTXDWLRQ UHPDLQ WKH VDPH UHJDUGOHVV ZKHWKHU WKH OHDGLQJ FRHIILFLHQW LV  RU QRW &DVH  7ZR GLVWLQFW UHDO URRWV :KHQ E í  DF !  7KH FKDUDFWHULVWLF SRO\QRPLDO KDYH WZR GLVWLQFW UHDO URRWV U U 7KH\ JLYH WZR GLVWLQFWÁ VROXWLRQV \ \ UW H  DQG UW H   7KHUHIRUH D JHQHUDO VROXWLRQ RI LV UW \ & \  & \ &H   &H U W  ,W LV WKDW HDV\ \  \  \ ([DPSOH  7KH FKDUDFWHULVWLF HTXDWLRQ LV U   U   U   U    WKH URRWV RI WKH SRO\QRPLDO DUH U í DQG í 7KH JHQHUDO VROXWLRQ LV WKHQ \ & H íW  & H íW 6XSSRVH WKHUH DUH LQLWLDO FRQGLWLRQV \   \  í $ XQLTXH SDUWLFXODU VROXWLRQ FDQ EH IRXQG E\ VROYLQJ IRU & DQG & XVLQJ WKH LQLWLDO FRQGLWLRQV )LUVW ZH QHHG WR FDOFXODWH \ í& H íW í & H íW WKHQ DSSO\ WKH LQLWLDO YDOXHV  \  & H   & H  í \  í & H  í  &  H  7KH VROXWLRQ LV & Á í DQG &  &  & í& í & \ íH íW   H íW :H VKDOO VHH WKH SUHFLVH PHDQLQJ RI GLVWLQFWQHVV LQ WKH QH[W VHFWLRQ )RU QRZ MXVW WKLQN WKDW WKH WZR VROXWLRQV DUH QRW FRQVWDQW PXOWLSOHV RI HDFK RWKHU 4XHVWLRQ 6XSSRVH WKH LQLWLDO FRQGLWLRQV DUH LQVWHDG \   \  í +RZ ZRXOG WKH QHZ W FKDQJH WKH SDUWLFXODU VROXWLRQ" $SSO\ WKH LQLWLDO FRQGLWLRQV DV EHIRUH DQG ZH VHH WKHUH LV D OLWWOH FRPSOLFDWLRQ 1DPHO\ WKH VLPXOWDQHRXV V\VWHP RI  HTXDWLRQV WKDW ZH KDYH WR VROYH LQ RUGHU WR ILQG & DQG & QRZ FRPHV ZLWK UDWKHU LQFRQYHQLHQW LUUDWLRQDO FRHIILFLHQWV  \  & H í  & H í í \  í& H í í & H í :LWK VRPH JRRG ERRNNHHSLQJ V\VWHPV OLNH WKLV FDQ EH VROYHG WKH XVXDO ZD\ +RZHYHU WKHUH LV DQ HDVLHU PHWKRG WR VLPSOLI\ WKH LQFRQYHQLHQW FRHIILFLHQWV 7KH LGHD LV WUDQVODWLRQ RU WLPHVKLIW  :KDW ZH ZLOO GR LV WR ILUVW FRQVWUXFW D QHZ FRRUGLQDWH D[LV VD\ D[LV 7KH WZR FRRUGLQDWHD[HV DUH UHODWHG E\ WKH HTXDWLRQ W í W 7KHUHIRUH ZKHQ W W  WKDW LV WKH LQLWLDO WYDOXH W EHFRPHV WKH QHZ RULJLQ ,Q RWKHU ZRUGV ZH WUDQVODWH RU WLPHVKLIW W D[LV E\ W XQLWV WR PDNH LW D[LV ,Q WKLV H[DPSOH ZH ZLOO DFFRUGLQJO\ VHW W í  7KH LPPHGLDWH HIIHFW LV WKDW LW PDNHV WKH LQLWLDO FRQGLWLRQV WR EH EDFN DW  \   \  í ZLWK UHVSHFW WR WKH QHZ FRRUGLQDWH :H WKHQ VROYH WKH WUDQVODWHG V\VWHP RI  HTXDWLRQV WR ILQG & DQG & :KDW ZH JHW LV WKH VLPSOHU V\VWHP  \  & H   & H  í \  í & H  í  &  H  &  & í& í & $V ZH KDYH VHHQ RQ WKH SUHYLRXV SDJH WKH VROXWLRQ LV & í DQG &  +HQFH WKH VROXWLRQ LQ WKH QHZ FRRUGLQDWH V\VWHP LV \ íH í   H í  /DVWO\ VLQFH WKLV VROXWLRQ LV LQ WHUPV RI  EXW WKH RULJLQDO SUREOHP ZDV LQ WHUPV RI W ZH VKRXOG FRQYHUW LW EDFN WR WKH RULJLQDO FRQWH[W 7KLV FRQYHUVLRQ LV HDVLO\ DFKLHYHG XVLQJ WKH WUDQVODWLRQ IRUPXOD XVHG HDUOLHU W í W W í  %\ UHSODFLQJ HYHU\ RFFXUUHQFH RI E\ W í  LQ WKH VROXWLRQ ZH REWDLQ WKH VROXWLRQ LQ LWV SURSHU LQGHSHQGHQW YDULDEOH W \ W íH í W í    H í W í   ([DPSOH &RQVLGHU WKH VROXWLRQ \ W RI WKH LQLWLDO YDOXH SUREOHP \ í \ í \  \   \   'HSHQGLQJ RQ WKH YDOXH RI  DV W ’ WKHUH DUH  SRVVLEOH EHKDYLRUV RI \ W  ([SOLFLWO\ GHWHUPLQH WKH SRVVLEOH EHKDYLRUV DQG WKH UHVSHFWLYH LQLWLDO YDOXH DVVRFLDWHG ZLWK HDFK EHKDYLRU 7KH FKDUDFWHULVWLF HTXDWLRQ LV U í  U í  U   U í  DUH U í DQG  7KH JHQHUDO VROXWLRQ LV WKHQ \  ,WV URRWV & H íW  & H W 1RWLFH WKDW WKH ORQJWHUP EHKDYLRU RI WKH VROXWLRQ LV GHSHQGHQW RQ WKH FRHIILFLHQW & RQO\ VLQFH WKH & H íW WHUP WHQGV WR  DV W ’ UHJDUGOHVV RI WKH YDOXH RI & 6ROYLQJ IRU & LQ WHUPV RI  ZH JHW \  \    &   & í&  &  &  & D S   1RZ LI & !  WKHQ \ WHQGV WR ’ DV W ’ 7KLV ZRXOG KDSSHQ ZKHQ ! í  ,I &  WKHQ \ WHQGV WR  DV W ’ 7KLV ZRXOG KDSSHQ ZKHQ í  /DVWO\ LI &   WKHQ \ WHQGV WR í’ DV W ’ 7KLV ZRXOG KDSSHQ ZKHQ  í  ,Q VXPPDU\ :KHQ !í  :KHQ í :KHQ í  OLP \ W & !  W of & f W of  &   OLP \ W   OLP \ W f  W of 7KH ([LVWHQFH DQG 8QLTXHQHVV RI WKH VROXWLRQ RI D VHFRQG RUGHU OLQHDU HTXDWLRQ LQLWLDO YDOXH SUREOHP $ VLEOLQJ WKHRUHP RI WKH ILUVW RUGHU OLQHDU HTXDWLRQ ([LVWHQFH DQG 8QLTXHQHVV 7KHRUHP«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c  \ 7KH VWDQGDUG IRUP LV \ cc  W W   VLQ W  \  í W    DQG W W  7KH GLVFRQWLQXLWLHV RI S T DQG J DUH W í  r  r  r « 7KH ODUJHVW LQWHUYDO WKDW FRQWDLQV W  EXW QRQH RI WKH GLVFRQWLQXLWLHV LV WKHUHIRUH   E   W  \ cc  OQ W   \ c  FRV W \ 7KH VWDQGDUG IRUP LV \ cc  OQ W    \c    \  FRV W  \  \     S W LV RQO\   W   W GHILQHG DQG LV FRQWLQXRXV RQ WKH LQWHUYDO í   DQG VLPLODUO\ T W LV RQO\ FRQWLQXRXVO\ GHILQHG RQ WKH LQWHUYDO í   J W LV FRQWLQXRXV HYHU\ZKHUH &RPELQLQJ WKHP ZH VHH WKDW S T DQG J KDYH GLVFRQWLQXLWLHV DW DQ\ W VXFK WKDW W ” í RU W •  7KDW LV WKH\ DUH DOO FRQWLQXRXV RQO\ RQ WKH LQWHUYDO í   6LQFH WKDW LQWHUYDO FRQWDLQV W  LW PXVW EH WKH ODUJHVW LQWHUYDO RQ ZKLFK WKH VROXWLRQ LV JXDUDQWHHG WR H[LVW XQLTXHO\ 7KHUHIRUH WKH DQVZHU LV í  ,QLWLDO 9DOXH 3UREOHP YV %RXQGDU\ 9DOXH 3UREOHP ,W PLJKW VHHP WKDW WKHUH DUH PRUH WKDQ RQH ZD\V WR SUHVHQW WKH LQLWLDO FRQGLWLRQV RI D VHFRQG RUGHU HTXDWLRQ ,QVWHDG RI ORFDWLQJ ERWK LQLWLDO FRQGLWLRQV \ W \ DQG \ W \  DW WKH VDPH SRLQW W FRXOGQ¶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† ZKHUH \ DQG \ DUH WZR ³GLVWLQFW´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c \ & \ W   &  \  W  c & \ W   &  \ c W   )URP OLQHDU DOJHEUD ZH NQRZ WKDW IRU WKH DERYH V\VWHP WR DOZD\V KDYH D XQLTXH VROXWLRQ & & IRU DQ\ LQLWLDO YDOXHV \ DQG \  WKH FRHIILFLHQW PDWUL[ RI WKH V\VWHP PXVW EH LQYHUWLEOH RU HTXLYDOHQWO\ WKH GHWHUPLQDQW RI WKH FRHIILFLHQW PDWUL[ PXVW EH QRQ]HUR  7KDW LV § \ W GHW ¨   ¨ \ c W © \  W  · ¸ c \  W  ¸ ¹ c c \ W  \  W   \ W  \  W  z  7KLV GHWHUPLQDQW DERYH LV FDOOHG WKH :URQVNLDQ RU WKH :URQVNLDQ GHWHUPLQDQW ,W LV D IXQFWLRQ RI W DV ZHOO GHQRWHG : \ \ W  DQG LV JLYHQ E\ WKH H[SUHVVLRQ : \ \ W † \ \  í \  \   7KH H[SUHVVLRQ \ & \  & \ LV FDOOHG D OLQHDU FRPELQDWLRQ RI WKH IXQFWLRQV \ DQG \ %\ QRQ]HUR LW PHDQV WKDW WKH :URQVNLDQ LV QRW WKH FRQVWDQW ]HUR IXQFWLRQ ,I : \ \ W 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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 U EH DQ\ WZR GLIIHUHQW § H U W GHW¨ U W ¨U H © U W U W U  U H  H UW H · ¸ UW UH  ¸ ¹ z U W U W UH  H U W U W  UH  H IRU DOO W 7KHUHIRUH DQ\ WZR GLIIHUHQW H[SRQHQWLDOIXQFWLRQ VROXWLRQV RI D VHFRQG RUGHU KRPRJHQHRXV OLQHDU HTXDWLRQ DV WKRVH IRXQG XVLQJ LWV FKDUDFWHULVWLF HTXDWLRQ DUH DOZD\V OLQHDUO\ LQGHSHQGHQW WKXV WKH\ ZLOO DOZD\V JLYH D JHQHUDO VROXWLRQ %HWWHU \HW LQ WKLV FDVH VLQFH WKH :URQVNLDQ LV QHYHU ]HUR IRU DOO UHDO QXPEHUV D XQLTXH VROXWLRQ FDQ DOZD\V EH IRXQG $EHO¶V 7KHRUHP ,I \ DQG \ DUH DQ\ WZR VROXWLRQV RI WKH HTXDWLRQ \  S W \  T W \  ZKHUH S DQG T DUH FRQWLQXRXV RQ DQ RSHQ LQWHUYDO , 7KHQ WKH :URQVNLDQ : \ \ W LV JLYHQ E\ : \  \ W  S W GW &H ³  ZKHUH & LV D FRQVWDQW WKDW GHSHQGV RQ \ DQG \ EXW QRW RQ W )XUWKHU : \ \ W LV HLWKHU ]HUR IRU DOO W LQ , LI &  RU HOVH LV QHYHU ]HUR LQ , LI &    &DVH  7ZR FRPSOH[ FRQMXJDWH URRWV :KHQ E í  DF   WKH FKDUDFWHULVWLF SRO\QRPLDO KDV WZR FRPSOH[ URRWV ZKLFK DUH FRQMXJDWHV U  L DQG U íL  DUH UHDO QXPEHUV UW !   $V EHIRUH WKH\ JLYH WZR OLQHDUO\ LQGHSHQGHQW VROXWLRQV \ \ UW H   &RQVHTXHQWO\ WKH OLQHDU FRPELQDWLRQ \ UW H  DQG UW &H   &H  ZLOO EH D JHQHUDO VROXWLRQ $W WKLV MXQFWXUH \RX PLJKW KDYH WKLV TXHVWLRQ ³EXW DUHQ¶W U DQG U FRPSOH[ QXPEHUV ZKDW ZRXOG EHFRPH RI WKH H[SRQHQWLDO IXQFWLRQ ZLWK D FRPSOH[ QXPEHU H[SRQHQW"´ 7KH DQVZHU WR WKDW TXHVWLRQ LV JLYHQ E\ WKH (XOHU¶V IRUPXOD (XOHU¶V IRUPXOD H L )RU DQ\ UHDO QXPEHU  FRV  L VLQ  +HQFH ZKHQ U LV D FRPSOH[ QXPEHU  L WKH H[SRQHQWLDO IXQFWLRQ H UW EHFRPHV H UW H 6LPLODUO\ ZKHQ U  L W H WH LW H W FRV W  L VLQ W í L  H UW EHFRPHV H í L W H W H í L W H W FRV í W  L VLQ í W H W FRV W í L VLQ W +HQFH WKH JHQHUDO VROXWLRQ IRXQG DERYH LV WKHQ \ & H W FRV W  L VLQ W  & H W FRV W í L VLQ W +RZHYHU WKLV JHQHUDO VROXWLRQ LV D FRPSOH[YDOXHG IXQFWLRQ PHDQLQJ WKDW JLYHQ D UHDO QXPEHU W WKH YDOXH RI WKH IXQFWLRQ \ W FRXOG EH FRPSOH[  ,W UHSUHVHQWV WKH JHQHUDO IRUP RI DOO SDUWLFXODU VROXWLRQV ZLWK HLWKHU UHDO RU FRPSOH[ QXPEHU FRHIILFLHQWV :KDW ZH VHHN KHUH LQVWHDG LV D UHDOYDOXHG H[SUHVVLRQ WKDW JLYHV RQO\ WKH VHW RI DOO SDUWLFXODU VROXWLRQV ZLWK UHDO QXPEHU FRHIILFLHQWV RQO\ ,Q RWKHU ZRUGV ZH ZRXOG OLNH WR ³ILOWHU RXW´ DOO IXQFWLRQV FRQWDLQLQJ FRHIILFLHQWV ZLWK DQ LPDJLQDU\ SDUW WKDW VDWLVI\ WKH JLYHQ GLIIHUHQWLDO HTXDWLRQ NHHSLQJ RQO\ WKRVH ZKRVH FRHIILFLHQWV DUH UHDO QXPEHUV 'HILQH X W Y W H W FRV W H W V LQ W ,W LV HDV\ WR YHULI\ WKDW ERWK X DQG Y VDWLVI\ WKH GLIIHUHQWLDO HTXDWLRQ RQH ZD\ WR VHH WKLV LV WR REVHUYH WKDW X FDQ EH REWDLQ IURP WKH FRPSOH[YDOXHG JHQHUDO VROXWLRQ E\ VHWWLQJ & &  DQG Y FDQ EH REWDLQHG VLPLODUO\ E\ VHWWLQJ & L DQG & íL  7KHLU :URQVNLDQ LV : X Y H  W LV QHYHU ]HUR 7KHUHIRUH WKH IXQFWLRQV X DQG Y DUH OLQHDUO\ LQGHSHQGHQW VROXWLRQV RI WKH HTXDWLRQ 7KH\ IRUP D SDLU RI UHDOYDOXHG IXQGDPHQWDO VROXWLRQV DQG WKH OLQHDU FRPELQDWLRQ LV D GHVLUHG UHDOYDOXHG JHQHUDO VROXWLRQ \ :KHQ U r L SRO\QRPLDO & H W FRV W  & H W VLQ W !  DUH WZR FRPSOH[ URRWV RI WKH FKDUDFWHULVWLF \  \ ([DPSOH $QVZHU \ ([DPSOH  &  FR V  W  &  V LQ  W \  \  \  \   \   7KH FKDUDFWHULVWLF HTXDWLRQ LV U  U    ZKLFK KDV VROXWLRQV U í r L 6R í DQG  7KHUHIRUH WKH JHQHUDO VROXWLRQ LV \ &  H í W FR V  W  &  H í W V LQ  W $SSO\ WKH LQLWLDO FRQGLWLRQV WR ILQG WKDW & \  DQG &  +HQFH  H í W FR V  W   H í W V LQ  W 4XHVWLRQ :KDW ZRXOG WKH VROXWLRQ EH LI WKH LQLWLDO FRQGLWLRQV DUH \   DQG \   LQVWHDG" $QVZHU \ H í W í  FRV  W í   H í W í  VLQ  W í  &DVH  2QH UHSHDWHG UHDO URRW :KHQ E í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cc  \ c  \   ZKHUH S W  &RPSXWH WKH :URQVNLDQ D D D WZR GLIIHUHQW ZD\V : \  \ § H UW GHW¨ UW ¨ UH © : \  \ &H ³ DQG  S W GW \ · ¸ c¹ \ ¸ &H c H UW \  UH UW \  U  E ³ D GW &H E W D  E D &   %\ WKH $EHO¶V 7KHRUHP WKH IDFW &   JXDUDQWHHV WKDW \ DQG \ DUH JRLQJ WR EH OLQHDUO\ LQGHSHQGHQW 1RZ ZH KDYH WZR H[SUHVVLRQV IRU WKH :URQVNLDQ RI WKH VDPH SDLU RI VROXWLRQV 7KH WZR H[SUHVVLRQV PXVW EH HTXDO H E W D E E D W c \  H \ D &H E W D  &   7KLV LV D ILUVW RUGHU OLQHDU GLIIHUHQWLDO HTXDWLRQ ZLWK \ DV WKH XQNQRZQ 3XW LW LQWR LWV VWDQGDUG IRUP DQG VROYH E\ WKH LQWHJUDWLQJ IDFWRU PHWKRG E c \  \ D P 7KH LQWHJUDWLQJ IDFWRU LV &H E W D E H ³  D GW H E W D  +HQFH  \ H & WH E W D E W D ³H E W D  & H &H E W D E W D GW H E W D ³ & GW H E W D & W  & & WH U W  &H U W $Q\ VXFK D IXQFWLRQ ZRXOG EH D VHFRQG OLQHDUO\ LQGHSHQGHQW VROXWLRQ RI WKH GLIIHUHQWLDO HTXDWLRQ :H MXVW QHHG RQH LQVWDQFH RI VXFK D IXQFWLRQ 7KH RQO\ FRQGLWLRQ IRU WKH FRHIILFLHQWV LQ WKH DERYH H[SUHVVLRQ LV &   3LFN VD\ &  DQG &  ZRXOG ZRUN QLFHO\ 7KXV \ W H UW 7KHUHIRUH WKH JHQHUDO VROXWLRQ LQ WKH FDVH RI D UHSHDWHG UHDO URRW U LV \ & H UW  & W H UW ([DPSOH \ í \  \  \   \  7KH FKDUDFWHULVWLF HTXDWLRQ LV U í U   U í    ZKLFK KDV VROXWLRQ U  UHSHDWHG  7KXV WKH JHQHUDO VROXWLRQ LV \ & H W  & W H W 'LIIHUHQWLDWH \ & H W  & W H W  H W  $SSO\ WKH LQLWLDO FRQGLWLRQV WR ILQG WKDW & \  H W í W H W  DQG & í  6XPPDU\ *LYHQ D VHFRQG RUGHU OLQHDU HTXDWLRQ ZLWK FRQVWDQW FRHIILFLHQWV D\  E\  F\  D    6ROYH LWV FKDUDFWHULVWLF HTXDWLRQ D U  E U  F  7KH JHQHUDO VROXWLRQ GHSHQGV RQ WKH W\SH RI URRWV REWDLQHG XVH WKH TXDGUDWLF IRUPXOD WR ILQG WKH URRWV LI \RX DUH XQDEOH WR IDFWRU WKH SRO\QRPLDO   :KHQ E í  DF !  WKHUH DUH WZR GLVWLQFW UHDO URRWV U U UW UW \ &H   &H    :KHQ E í  DF   WKHUH DUH WZR FRPSOH[ FRQMXJDWH URRWV U 7KHQ \ & H W FRV W  & H W VLQ W  :KHQ E í  DF  WKHUH LV RQH UHSHDWHG UHDO URRW U 7KHQ \ & H UW  & W H UW r L ([HUFLVHV D\  E\  F\ E  LV D VROXWLRQ RI WKH HTXDWLRQ D W H UW U  9HULI\ WKDW \  LI E í  DF  DQG LW LV QRW D VROXWLRQ LI E í  DF    í  )RU HDFK RI WKH IROORZLQJ HTXDWLRQV D ILQG LWV JHQHUDO VROXWLRQ E ILQG WKH SDUWLFXODU VROXWLRQ VDWLVI\LQJ WKH LQLWLDO FRQGLWLRQV \   \  í DQG F ILQG WKH OLPLW DV W ’ RI WKH VROXWLRQ IRXQG LQ E   \  \  \   \ í \  \   \ í \  \  \  \ í \  \ í \  \  \  \  \  \ í \  \  \  \   \ í \  \        ±  6ROYH HDFK LQLWLDO YDOXH SUREOHP  \  \  \  \   \    \ í \  \ í  \  \  \  \  \  \ í \  \  \ í  \ í \  í \   \   \   \  í \    í í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§\ GHW¨  ¨ \c © \Y W · ¸ c \Y W  \Yc W ¸ ¹  c c \ \Y W  \ Yc W  \ \Y W %\ WKH $EHO¶V 7KHRUHP : \  \  S W GW &H ³  ZKHUH &   7KH IDFW WKDW &   LV LPSRUWDQW EHFDXVH LW JXDUDQWHHV WKH OLQHDU LQGHSHQGHQFH RI \ DQG \  \ Yc W 7KH WZR H[SUHVVLRQV FRPSXWHG DERYH DUH WKH :URQVNLDQ RI WKH VDPH WZR IXQFWLRQV WKHUHIRUH WKH WZR H[SUHVVLRQV PXVW EH WKH VDPH (TXDWH WKHP  \ Yc W Yc W 7KHUHIRUH  S W GW &H ³   S W GW H³ &  \  ,QWHJUDWH WKH ULJKWKDQG VLGH WR ILQG Y W  7KHQ ILQG \ 7KH JHQHUDO VROXWLRQ LV VWLOO RI FRXUVH LQ WKH IRUP \ 7KHUHIRUH \ &  \  &  \ &   \ Y W  &  \  &  \  & \  & \ Y W  1RWH ,W LV DFWXDOO\ QRW QHFHVVDU\ WR DVVXPH WKDW \ \ Y W  $OWKRXJK GRLQJ VR PDNHV WKH UHVXOWLQJ ILUVW RUGHU GLIIHUHQWLDO HTXDWLRQ HDVLHU WR VROYH ([DPSOH ,I LW LV NQRZQ WKDW \ W LV D VROXWLRQ RI W \ í W W   \  W   \  W !  )LQG LWV JHQHUDO VROXWLRQ 5HZULWH WKH HTXDWLRQ LQWR WKH VWDQGDUG IRUP \ cc  ,GHQWLI\ S W  W W \c   \ W W W  /HW \ W  \ Y W W Y W  W Y W · §W   GHW¨ ¨ Y W  W Yc W ¸ W Y W  W Yc W  W Y W W Yc W  ¸ © ¹ : \  \ DQG : \  \  &H ³ W  GW W § &H · ³ ¨  W ¸ GW © ¹ & H W  OQ W  & W HW  ZKHUH &   W  Yc & W  H W (TXDWLQJ ERWK SDUWV Y &KRRVH & &HW Y  DQG &  Y & H W  & H W 7KHUHIRUH \ 7KH JHQHUDO VROXWLRQ LV \ &  \  &  \ & W  & W H W \ Y W W H W 6HFRQG 2UGHU /LQHDU 1RQKRPRJHQHRXV 'LIIHUHQWLDO (TXDWLRQV 0HWKRG RI 8QGHWHUPLQHG &RHIILFLHQWV :H ZLOO QRZ WXUQ RXU DWWHQWLRQ WR QRQKRPRJHQHRXV VHFRQG RUGHU OLQHDU HTXDWLRQV HTXDWLRQV ZLWK WKH VWDQGDUG IRUP \  S W \  T W \ J W  J W   (DFK VXFK QRQKRPRJHQHRXV HTXDWLRQ KDV D FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ \  S W \  T W \  1RWH WKDW WKH WZR HTXDWLRQV KDYH WKH VDPH OHIWKDQG VLGH LV MXVW WKH KRPRJHQHRXV YHUVLRQ RI  ZLWK J W  :H ZLOO IRFXV RXU DWWHQWLRQ WR WKH VLPSOHU WRSLF RI QRQKRPRJHQHRXV VHFRQG RUGHU OLQHDU HTXDWLRQV ZLWK FRQVWDQW FRHIILFLHQWV D\  E\  F\ J W  :KHUH D E DQG F DUH FRQVWDQWV D   DQG J W   ,W KDV D FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ D \  E \  F \  6ROXWLRQ RI WKH QRQKRPRJHQHRXV OLQHDU HTXDWLRQV ,W FDQ EH YHULI\ HDVLO\ WKDW WKH GLIIHUHQFH \ < í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³HGXFDWHG JXHVVHV´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í \ í \ H W 7KH FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ \ í \ í \  KDV FKDUDFWHULVWLF HTXDWLRQ U í U í  U   U í   6R WKH íW W FRPSOHPHQWDU\ VROXWLRQ LV \F & H  & H  7KH QRQKRPRJHQHRXV HTXDWLRQ KDV J W HW ,W LV DQ H[SRQHQWLDO IXQFWLRQ ZKLFK GRHV QRW FKDQJH IRUP DIWHU GLIIHUHQWLDWLRQ DQ H[SRQHQWLDO IXQFWLRQ¶V GHULYDWLYH ZLOO UHPDLQ DQ H[SRQHQWLDO IXQFWLRQ ZLWK WKH VDPH H[SRQHQW DOWKRXJK LWV FRHIILFLHQW PLJKW FKDQJH GXH WR WKH HIIHFW RI WKH &KDLQ 5XOH  7KHUHIRUH ZH FDQ YHU\ UHDVRQDEO\ H[SHFW WKDW < W LV LQ WKH IRUP $ HW IRU VRPH XQNQRZQ FRHIILFLHQW $ 2XU MRE LV WR ILQG WKLV DV \HW XQGHWHUPLQHG FRHIILFLHQW W W W /HW < $ H  WKHQ < $ H  DQG < $ H  6XEVWLWXWH WKHP EDFN LQWR WKH RULJLQDO GLIIHUHQWLDO HTXDWLRQ $ HW í  $ HW í  $ HW í  $ H W $ H W H W í   +HQFH < W   W H  7KHUHIRUH \ \F  <  &H W  &  H W  H W   7KLQJ WR UHPHPEHU :KHQ DQ H[SRQHQWLDO IXQFWLRQ DSSHDUV LQ J W  XVH DQ H[SRQHQWLDO IXQFWLRQ RI WKH VDPH H[SRQHQW IRU < \ í \ í \ ([DPSOH  W   W í  7KH FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ LV VWLOO \ í \ í \  íW W 7KHUHIRUH WKH FRPSOHPHQWDU\ VROXWLRQ UHPDLQV \F & H  & H  1RZ J W W  W í  ,W LV D GHJUHH  LH TXDGUDWLF SRO\QRPLDO 6LQFH SRO\QRPLDOV OLNH H[SRQHQWLDO IXQFWLRQV GR QRW FKDQJH IRUP DIWHU GLIIHUHQWLDWLRQ WKH GHULYDWLYH RI D SRO\QRPLDO LV MXVW DQRWKHU SRO\QRPLDO RI RQH GHJUHH OHVV XQWLO LW HYHQWXDOO\ UHDFKHV ]HUR  :H H[SHFW WKDW < W ZLOO WKHUHIRUH EH D SRO\QRPLDO RI WKH VDPH GHJUHH DV WKDW RI J W  :K\ ZLOO WKHLU GHJUHHV EH WKH VDPH"  6R ZH ZLOO OHW < EH D JHQHULF TXDGUDWLF SRO\QRPLDO < $ W  % W  & ,W IROORZV < $ W  % DQG < $ 6XEVWLWXWH WKHP LQWR WKH HTXDWLRQ $ í  $ W  % í  $ W  % W  & W  W í  í $ W  í $ í % W  $ í % í & W  W í  7KH FRUUHVSRQGLQJ WHUPV RQ ERWK VLGHV VKRXOG KDYH WKH VDPH FRHIILFLHQWV WKHUHIRUH HTXDWLQJ WKH FRHIILFLHQWV RI OLNH WHUPV W  W   í $  í$ í % í $ í % í & 7KHUHIRUH < í W   DQG \ $ % & \F  < í   &  Hí W  &  H W í W     7KLQJ WR UHPHPEHU :KHQ D SRO\QRPLDO DSSHDUV LQ J W  XVH D JHQHULF SRO\QRPLDO RI WKH VDPH GHJUHH IRU < 7KDW LV < $Q WQ  $Qí WQí  «  $ W  $ 1RWH WKDW LI J W LV D QRQ]HUR FRQVWDQW LW LV FRQVLGHUHG D SRO\QRPLDO RI GHJUHH  DQG < ZRXOG WKHUHIRUH DOVR EH D JHQHULF SRO\QRPLDO RI GHJUHH  7KDW LV < LV DQ DUELWUDU\ QRQ]HUR FRQVWDQW < $ 5HFDOO WKDW WKH GHJUHH RI D SRO\QRPLDO LV WKH KLJKHVW SRZHU WKDW DSSHDUV 7KHUHIRUH WKH UXOH FDQ EH VWDWHG D OLWWOH GLIIHUHQWO\ WR VD\ WKDW ³ORRN IRU WKH KLJKHVW SRZHU RI W LQ J W  WKHQ OLVW LW DQG DOO WKH ORZHU SRZHUV GRZQ WR WKH FRQVWDQW WHUP LQ < \ í \ í \ ([DPSOH FRV W $JDLQ WKH VDPH FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ DV WKH SUHYLRXV H[DPSOHV PHDQV WKDW \F & HíW  & HW DV EHIRUH 7KH QRQKRPRJHQHRXV WHUP LV J W FRV W  &RVLQH DQG VLQH IXQFWLRQV GR FKDQJH IRUP VOLJKWO\ ZKHQ GLIIHUHQWLDWHG EXW WKH SDWWHUQ LV VLPSOH SUHGLFWDEOH DQG UHSHWLWLYH WKHLU UHVSHFWLYH IRUPV MXVW FKDQJH WR HDFK RWKHU¶V &RQVHTXHQWO\ ZH VKRXOG FKRRVH WKH IRUP < $ FRV W  % VLQ W  :K\ GR ZH FKRRVH WR HPSOR\ ERWK FRVLQH DQG VLQH" 6XEVWLWXWH < < í$ VLQ W  % FRV W  DQG < í$ FRV W í % VLQ W LQWR WKH HTXDWLRQ í$ FRV W í % VLQ W í  í$ VLQ W  % FRV W í  $ FRV W  % VLQ W FRV W í$ í % í $ FRV W  í%   $ í % VLQ W í$ í % FRV W   $ í % VLQ W FRV W FRV W   VLQ W &RPSDUH WKH FRHIILFLHQWV FRV W  VLQ W  7KHUHIRUH < \   í$ í %  $ í % $ % í   í     FRV W  VLQ W  DQG   &H W  &  H W    FRV W  VLQ W   7KLQJ WR UHPHPEHU :KHQ HLWKHU FRVLQH RU VLQH DSSHDUV LQ J W  ERWK FRVLQH DQG VLQH RI WKH VDPH IUHTXHQF\ PXVW DSSHDU LQ < :KHQ J W LV D VXP RI VHYHUDO WHUPV :KHQ J W LV D VXP RI VHYHUDO IXQFWLRQV J W J W  J W  «  JQ W  ZH FDQ EUHDN WKH HTXDWLRQ LQWR Q SDUWV DQG VROYH WKHP VHSDUDWHO\ *LYHQ \  S W \  T W \ J W  J W  «  JQ W ZH FKDQJH LW LQWR \  S W \  T W \ J W \  S W \  T W \   \  S W \  T W \ J W JQ W  6ROYH WKHP LQGLYLGXDOO\ WR ILQG UHVSHFWLYH SDUWLFXODU VROXWLRQV < < «  <Q 7KHQ DGG XS WKHP WR JHW < <  <  «  <Q \ í \ í \ ([DPSOH HW  W  W í   FRV W 6ROYH HDFK RI WKH VXESDUWV \ í \ í \ H W < W   W H  \ í \ í \  W  W í  < W í W   \ í \ í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í \ í \ W  W í  \  íW W  \   )LUVW ILQG WKH JHQHUDO VROXWLRQ \ & H  & H í W   7KHQ XVH WKH LQLWLDO FRQGLWLRQV WR ILQG WKDW &  DQG &  7KH SDUWLFXODU VROXWLRQ LV \  Hí W  H W í W     í $ SRVVLEOH JOLWFK" 7KHUH LV D FRPSOLFDWLRQ WKDW RFFXUV XQGHU D FHUWDLQ FLUFXPVWDQFH« ([DPSOH \ í \ í \  HW 7KH ROG QHZV LV WKDW \F & HíW  & HW 6LQFH J W HW ZH VKRXOG EH DEOH WR XVH WKH IRUP < $ HW MXVW OLNH LQ WKH ILUVW H[DPSOH ULJKW" $ HW LQWR WKH GLIIHUHQWLDO %XW LI ZH VXEVWLWXWH < < $ HW DQG < HTXDWLRQ DQG VLPSOLI\ ZH ZRXOG JHW WKH HTXDWLRQ  HW 7KDW PHDQV WKHUH LV QR VROXWLRQ IRU $ 2XU PHWKRG WKDW KDV ZRUNHG ZHOO WKXV IDU VHHPV WR KDYH IDLOHG 7KH VDPH RXWFRPH DQ LQDELOLW\ WR ILQG $ DOVR KDSSHQV ZKHQ J W LV D PXOWLSOH RI HíW %XW IRU DQ\ RWKHU H[SRQHQW RXU FKRLFH RI WKH IRUP IRU < ZRUNV :KDW LV VR VSHFLDO DERXW WKHVH WZR SDUWLFXODU H[SRQHQWLDO IXQFWLRQV HW DQG Hí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í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í  $W HW  $ W HW   $ W HW $ HW   í  $W HW   í    $ W HW $ HW $ +HQFH < W 7KHUHIRUH \   W WH   HW H W HW   &H W  &  WH W  W  H W  2XU ³FXUH´ KDV ZRUNHG  6LQFH D VHFRQG RUGHU OLQHDU HTXDWLRQ¶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í \ í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¶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í HW FRV W 7KH FRPSOHPHQWDU\ VROXWLRQ LV \F & FRV W  & VLQ W  /HW¶V EUHDN XS J W LQWR  SDUWV DQG ZRUN RQ WKHP LQGLYLGXDOO\ J W W VLQ W LV D SURGXFW RI D GHJUHH  SRO\QRPLDO DQG D VLQH IXQFWLRQ 7KHUHIRUH < VKRXOG EH D SURGXFW RI D JHQHULF GHJUHH  SRO\QRPLDO DQG ERWK FRVLQH DQG VLQH < $W  %W  &W  ' FRV W  (W  )W  *W  + VLQ W 7KH YDOLGLW\ RI WKH DERYH FKRLFH RI IRUP FDQ EH YHULILHG E\ RXU VHFRQG ORQJHU PHWKRG 1RWH WKDW WKH SURGXFW RI D GHJUHH  SRO\QRPLDO DQG ERWK FRVLQH DQG VLQH W  W  W   u FRV W  VLQ W FRQWDLQV  GLVWLQFW WHUPV OLVWHG EHORZ WFRV W WVLQ W WFRV W WVLQ W W FRV W W VLQ W FRV W VLQ W 1RZ LQVHUW  LQGHSHQGHQW XQGHWHUPLQHG FRHIILFLHQWV RQH IRU HDFK < $WFRV W  %WFRV W  &W FRV W  'FRV W  (WVLQ W  )WVLQ W  *W VLQ W  +VLQ W +RZHYHU WKHUH LV VWLOO RQH LPSRUWDQW GHWDLO WR FKHFN EHIRUH ZH FRXOG SXW WKH DERYH H[SUHVVLRQ GRZQ IRU < ,V WKHUH DQ\WKLQJ LQ WKH H[SUHVVLRQ WKDW LV VKDUHG ZLWK \F & FRV W  & VLQ W " $V ZH FDQ VHH WKHUH DUH ± ERWK WKH IRXUWK DQG WKH HLJKWK WHUPV 7KHUHIRUH ZH QHHG WR PXOWLSO\ HYHU\WKLQJ LQ WKLV HQWLUH H[SUHVVLRQ E\ W +HQFH < W $W  %W  &W  ' FRV W  W (W  )W  *W  + VLQ W $W  %W  &W  'W FRV W  (W  )W  *W  +W VLQ W  7KH VHFRQG KDOI RI J W LV J W íHW FRV W  ,W LV D SURGXFW RI DQ H[SRQHQWLDO IXQFWLRQ DQG FRVLQH 6R RXU FKRLFH RI IRUP IRU < VKRXOG EH D SURGXFW RI DQ H[SRQHQWLDO IXQFWLRQ ZLWK ERWK FRVLQH DQG VLQH < , HW FRV W  - HW VLQ W  7KHUH LV QR FRQIOLFW ZLWK WKH FRPSOHPHQWDU\ VROXWLRQ ± HYHQ WKRXJK ERWK FRV W DQG VLQ W DUH SUHVHQW ZLWKLQ ERWK \F DQG < WKH\ DSSHDU DORQH LQ \F EXW LQ SURGXFWV ZLWK HW LQ < PDNLQJ WKHP SDUWV RI FRPSOHWHO\ GLIIHUHQW IXQFWLRQV +HQFH WKLV LV WKH FRUUHFW FKRLFH )LQDOO\ WKH FRPSOHWH FKRLFH RI < LV WKH VXP RI < DQG < < <  < $W  %W  &W  'W FRV W  (W  )W  *W  +W VLQ W  , HW FRV W  - HW VLQ W  ([DPSOH \ í \  \ &RPSOHPHQWDU\ VROXWLRQ \F W HW í W VLQ W   & HW  & HW 7KH IRUP RI SDUWLFXODU VROXWLRQ LV < ([DPSOH $W  %W  &W HW  'W  ( FRV W  )W  * VLQ W  + W HíW í W HW FRV W  W í  \  \  \ &RPSOHPHQWDU\ VROXWLRQ \F & HíW  & W HíW 7KH IRUP RI SDUWLFXODU VROXWLRQ LV < $W  %W HíW  &W  'W  ( HW FRV W  )W  *W  + HW VLQ W  , W  - W  . ([DPSOH )LQG D VHFRQG RUGHU OLQHDU HTXDWLRQ ZLWK FRQVWDQW FRHIILFLHQWV ZKRVH JHQHUDO VROXWLRQ LV \ &HW  & HíW  W 7KH VROXWLRQ FRQWDLQV WKUHH SDUWV VR LW PXVW FRPH IURP D QRQKRPRJHQHRXV HTXDWLRQ 7KH FRPSOHPHQWDU\ SDUW RI WKH VROXWLRQ \F & HW  & HíW VXJJHVWV WKDW U  DQG U í DUH WKH WZR URRWV RI LWV FKDUDFWHULVWLF HTXDWLRQ +HQFH U ±  DQG U   DUH LWV WZR IDFWRUV 7KHUHIRUH WKH FKDUDFWHULVWLF HTXDWLRQ LV U ±  U   U  U ±  7KH FRUUHVSRQGLQJ KRPRJHQHRXV HTXDWLRQ LV DV D UHVXOW \  \ í \  +HQFH WKH QRQKRPRJHQHRXV HTXDWLRQ LV \  \ í \ J W  7KH QRQKRPRJHQHRXV SDUW J W UHVXOWV LQ WKH SDUWLFXODU VROXWLRQ < W $V ZHOO < W DQG <  7KHUHIRUH J W <  < í <    W ±  W   W ±  W 7KH HTXDWLRQ ZLWK WKH JLYHQ JHQHUDO VROXWLRQ LV WKHUHIRUH \  \ í \   W ±  W 7KH  5XOHVRI7KXPE RI WKH 0HWKRG RI 8QGHWHUPLQHG &RHIILFLHQWV  ,I DQ H[SRQHQWLDO IXQFWLRQ DSSHDUV LQ J W  WKH VWDUWLQJ FKRLFH IRU < W LV DQ H[SRQHQWLDO IXQFWLRQ RI WKH VDPH H[SRQHQW  ,I D SRO\QRPLDO DSSHDUV LQ J W  WKH VWDUWLQJ FKRLFH IRU < W LV D JHQHULF SRO\QRPLDO RI WKH VDPH GHJUHH  ,I HLWKHU FRVLQH RU VLQH DSSHDUV LQ J W  WKH VWDUWLQJ FKRLFH IRU < W QHHGV WR FRQWDLQ ERWK FRVLQH DQG VLQH RI WKH VDPH IUHTXHQF\  ,I J W LV D VXP RI VHYHUDO IXQFWLRQV J W J W  J W  «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«  JN W   )RU HDFK JL W  FKRRVH WKH IRUP RI LWV FRUUHVSRQGLQJ SDUWLFXODU VROXWLRQ <L W DFFRUGLQJ WR JL W <L W 3Q W W V $Q WQ  $Qí WQí  «  $ W  $ 3Q W HDW W V $Q WQ  $Qí WQí  «  $ W  $ HDW DW 3Q W H FRV W DQGRU 3Q W HDW VLQ W W V $Q WQ  $Qí WQí  «  $ HDW FRV W  V Q Qí W %Q W  %Qí W  «  % HDW VLQ W :KHUH V   RU  LV WKH PLQLPXP QXPEHU RI WLPHV WKH FKRLFH PXVW EH PXOWLSOLHG E\ W VR WKDW LW VKDUHV QR FRPPRQ WHUPV ZLWK \F 3Q W GHQRWHV D QWK GHJUHH SRO\QRPLDO ,I WKHUH LV QR SRZHU RI W SUHVHQW WKHQ Q  DQG 3 W & LV MXVW WKH FRQVWDQW FRHIILFLHQW ,I QR H[SRQHQWLDO WHUP LV SUHVHQW WKHQ VHW WKH H[SRQHQW D   < <   <   «  < N  7KH JHQHUDO VROXWLRQ LV \ \F  <   )LQDOO\ DSSO\ DQ\ LQLWLDO FRQGLWLRQV WR GHWHUPLQH WKH DV \HW XQNQRZQ FRHIILFLHQWV & DQG & LQ \F ...
View Full Document

Ask a homework question - tutors are online