=   TYYytermfunc Y[yyyterminalPthes 87Ytheta,O477YYyyythirdPtime PbtmsgPYytop TTY[byyyndQnew77no Y[nonintegY nonnegintY ;can'>nargPnaval[PQ'K4M4N4O47787TTTTYY[byyynavy[,PQ'K4M4N4O47787TTTTYY[byyyndQnew77no Y[nonintegY nonnegint Ybnonuniqu77nops'normal77note87YY[notic[numbbnumeric QYnw[PQqGF/F/*&%\"rGF/%\"sGF/F/F.F5*&%\"tGF.%\"uG F.F/7$/,$*$,&-%$sinG6#%\"xGF/#\"\"%\"\"$F/F.\"\"*-%\"fGFJ/,&*$,&%\"yGF /F-F5F.F/*$,&FKF/F1F5F.F/%\"FG" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. M urison\nAstronomical Applications Dept.\nU.S. Naval Observatory\n3450 \+ Massachusetts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 1 " " }}}} 8Ԍ tP*FH\"#C\"#;F/-%\"fGFJ/,.*$FKF3F/*&FKF/F.F/!\"#*$F.F3F/*$%\"yGF3F/* &FXF/F2F/FUF1F/%\"FG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "csq uare( M, [p*q,b^2,x,y,sin(x)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'MATRIXG6#7$7$/,&*$,&*$%\"bG\"\"#\"\"\"*&%\"aGF/%\"cGF/F/F.F/*$%\"dGF. !\"\"\"\"!,&*$,&*&%\"pGF/%\"T{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading y8{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 128 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 128 0 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 1C0 0 } 1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Add ress" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 Putilities,tmsgQutilities,csquareutilities,funcopsutilities,termfunc'ODE,diffcoeffsK4 ODE,type,diffM4 ODE,difforderN4 ODE,isdiffO4 ODE,varchange77ODE,indvarchange87ODETutilities,locationT ODE,depvarTODE,maxPDEorderT ODE,diffvarsYutilities,expansionY utilities[utilities,pulloutbutilities,applyfuncyutilities,cosfixyutilities,sinfixP utilitiesQ utilities utilities utilities'ODEK4ODEM4ODEN4ODEO4ODE77ODET utiitiesTODETODETODEY utilities[ utilitiesb utilitiesy utilitiesy utilities applyfuncbcosfixycsquareQdepvarTdiffK4 diffcoeffs' difforderM4diffvarsT expansionYfuncops indvarchange77isdiffN4locationT maxpdeorderTpullout[sinfixytermfunctmsgPtypeK4utilsY varchangeO4 M4TTYY[original'77Tout Y[param YY parameterY paramlistYpartPYcoordY correspond'cos O477TTYycosfixYyyycosin yycours[csquarQYycurrPdc[PQ'K4M4N4O47787TTTTYY[byyydebugPYyut[callP ; ;YLE "Hea ding 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Fon t 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 12 " \226 csquare()0argumpcallTdebugFdiffeqnequatfullindex"marc9name7nwpartquantitroutinnstepotoputilitmy polynomialQpolytestypossiblTpoweryprint PyprocPTyprocedurPbyprocnamPproducTprompt77pull[pullout [ ypurposOPQ'K4M4N4O477TTTTY[byyquantitTrath77read 87yrealy reconstruct87recovTrecover'repeat[replac77result[return'TTTTrhoO4y y rhs yyriemanS(PQ'K4M4N4O47787TTTT[byyyrootfuncy 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 12 " \226 pullout()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 260 7 "pullout" }{TEXT 259 66 "( expr::\{`+`,`^`,`*`\}, subexpr::\{name,function,`+`,`*`,`^`,list\} ) " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 39 "Selectively factor out a subexpression " }{TEXT 261 7 "su bexpr" }{TEXT -1 21 " from and expression " }{TEXT 262 4 "expr" } {TEXT -1 26 " (see the Examples below)." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 42 " Expression to selectively factor." }}{PARA 15 "" 0 "" {TEXT 258 10 "subexpr " }{TEXT -1 31 "Subexpression to factor out of " }{TEXT 263 4 "expr" }{TEXT -1 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "foo \+ := A*(B*x+C*B*y*x+z*B)^2+D;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fPtmsgQcsquarefuncops termfunc' diffcoeffsK4 typediffM4 difforderN4isdiffO4 varchange77 indvarchangeT locationTdepvarT maxpdeorderT diffvarsY expansionYutils[pulloutb applyfuncycosfixysinfixODE87 ODE,depvarTODE,diffcoeffs' ODE,difforderM4 ODE,diffvarsTODE,indvarchange77 ODE,isdiffN4ODE,maxPDEorderT ODE,type,diffK4 ODE,varchangeO4 utilitiesYutilities,applyfuncbutilities,cosfixyutilities,csquareQutilities,expansionYutilities,funcopsutilities,locationTutilities,pullout[utilities,sinfixyutilities,termfuncutilities,tmsgP'suppos77symbolic O477syntax[PQ'K4M4N4O47787TTTTYY[byyytablK4tan77TTtauYO4%yyteqn 7787termTYYytermfunc  ;argument[PQ'K4M4N4O47787TTTTYY[byyyarra#Q87YY yy y artificial77assum0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output " 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 9 " \226 tmsg()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 " Syntax" }}{PARA 0 "" 0 "" {TEXT 260 29 "tmsg( proc_name, msg <, t0> ) " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 31 "Print messages of the form\n " }{TEXT 20 35 " []: " }{TEXT -1 31 "\nto the terminal screen, wh ere " }{TEXT 20 14 "" }{TEXT -1 103 " is the current tim e minus the time stored in the optional third argument, with decimal p art truncated." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }} {PARA 15 "" 0 "" {TEXT 257 9 "proc_name" }{TEXT -1 55 " Name of the calling procedure. Usually 'procname'." }}{PARA 15 "" 0 "" {TEXT 258 7 "msg " }{TEXT -1 68 "Message string to be printed on the scre en. Must be a Maple string." }}{PARA 15 "" 0 "" {TEXT 259 6 "t0 " }{TEXT -1 16 "A fiducial time." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "SomeFunction := (msg)-> tmsg('procname',msg,args[2..nargs]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "time();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&1] *!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SomeFunction(`hi t here`);" }}{PARA 6 "" 1 "" {TEXT -1 26 "SomeFunction[95]: hi there" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "SomeFunction(`ho there`,tim e());" }}{PARA 6 "" 1 "" {TEXT -1 24 "SomeFunction[]: ho there" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 " " 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications De pt.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWashignton, \+ DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.m il" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/muri son/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "debug_print" 2 "utilities,debug_print" "" }{TEXT -1 1 " \+ " }}}} .nargs]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "time();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&1] *!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SomeFunction(`hi t here`);" }}{PARA 6 "" 1 "" {TEXT -1 26 "SomeFunction[95]: hi there" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "SomeFunction(`ho there`,tim e());" }}{PARA 6 "" 1 "" {TEXT -1 24 "SomeFunction[]: ho there" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 " " 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications De pt.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWashignton, \+ DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.m il" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/muri son/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " } {HYPERLNK 17 Q{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PST 5 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "termfunc" 2 "utilities,termfunc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "funcops" 2 "utilities,funcops" "" }{TEXT -1 1 " " }}}} RA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"DG\"\"\"%\"xGF&F&*&%\"BGF&,&%\"EG!\"\"*,%\"FGF&% \"AGF&F)F&F'\"\"#,(*&,&F&F&F'F&F&F'F,F&*&%\"CGF&%\"yGF&F&*(%\"zGF&F3F& F'F,F&F0F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\" -foo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval \+ Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }} {PARA 1.OPQ6Y''+'&t2?u2fK4:L4`M4eN4v.677>r<%s<TTRTzTwTTTbUYYD[B[bQbyyL{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1 " 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1,22 "(this is an alias for " }{TEXT -1 22 "`CombineTrig/optimize`" } {TEXT 263 1 ")" }}{PARA 278 "" 0 "" {TEXT -1 56 "common_factor( expr:: algebraic, varlist::\{list,string\} )" }}{PARA 287 "" 0 "" {HYPERLNK 17 "cosfix" 2 "utilities,cosfix" "" }{TEXT -1 57 "( expr::\{algebraic, array(1),`=`\}, var::\{algebraic,list\} )" }}{PARA 265 "" 0 "" {HYPERLNK 17 "csquare" 2 "utilities,csquare" "" }{TEXT -1 8 "( expr:: " }{TEXT 259 42 "\{algebraic,array,indexed,equation,numeric\}" }{TEXT 261 2 " \n" }{TEXT -1 17 " <, var::" }{TEXT 260 36 "\{name,set ,list,`+`,`*`,`^`,function\}" }{TEXT -1 3 "> )" }}{PARA 266 "" 0 "" {TEXT 258 51 "debug_print( pname, msg, verbosity_level <, expr> )" }} {PARA 279 "" 0 "" {TEXT -1 157 "expansion( expression::\{list,'array'( 1),`=`,algebraic\},\n params::\{string,list,function\}, expa nsion_order::integer, \n simplflag::integer )" }}{PARA 280 " " 0 "" {TEXT -1 213 "expansion2( expression::\{list,'array'(1),`=`,alg ebraic\},\n params1::\{str$hor" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU. S. Naval Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 2039 2" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }} {PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "funcops" 2 "utilities,funcops" "" }{TEXT -1 1 " " }}}} ^EXCHG 9again algebr ^SECTQ 9^{PARA 0 "" 0 "" {TEXT -1 38 "Again, this is not what we want. Use " }{TEXT 267 8 "termfunc" }{TEXT -1 1 ":" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "termfunc(\"\",simplify,\{x ^2-2*x*A+A^2+y^2+z^2=R1^2, x^2+2*x*A+A^2+y^2+z^2=R2^2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"BG\"\"\"*$%$R1|irG\"\"##!\"&F),&%\"xG !\"\"%\"AGF&F&\"\"$*(%\"CGF&*$%$R2|irGF)F*,&F-F.F/F.F&F0" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the AutPf{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Outpu t" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 0 128 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE " " -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 1O'P-QAa'u2qL4r.6#77Ur " 0 "" {MPLTEXT 1 0 16 "pul lout( \", x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"DG\"\"\"*&%\"AG F%,&*&%\"zGF%%\"BGF%F%*&%\"xGF%,&F+F%*(%\"CGF%F+F%%\"yGF%F%F%F%\"\"#F% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Note: " }{TEXT 264 6 "always" }{TEXT -1 18 " check the answer!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\"-foo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We can stack executions in a list. Pull out x then B." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pullout( foo, [x,B] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"D G\"\"\"*(%\"AGF%%\"BG\"\"#,&%\"zGF%*&%\"xGF%,&F%F%*&%\"CGF%%\"yGF%F%F% F%F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\"-foo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Notice that the order of executions is significant. Rep eat the pulling out of x and B, but in the order B then x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pullout( foo, [B,x] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"DG\"\"\"*(%\"AGF%%\"BG\"\"#,(%\"zGF%%\"xGF %*(%\"CGF%%\"yGF%F,F%F%F)F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "In this case, trying to pull out x had no effect." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Another sequence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "pullout( foo, B*x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"DG\"\"\"*&%\"AGF%,&*&%\"zGF%%\"BGF%F%*(%\"xGF%F+F%,&F%F%*&%\"C GF%%\"yGF%F%F%F%\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " pullout( \", B );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"DG\"\"\"*(% \"AGF%%\"BG\"\"#,&%\"zGF%*&%\"xGF%,&F%F%*&%\"CGF%%\"yGF%F%F%F%F)F%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\"-foo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "We could, of course, have done this in one step." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pullout( foo, [B*x,B] );" }}{PARA 11 "" " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "termfunc( expr, factor );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*(%\"BG\"\"\",&%\"xGF&%\"AG!\"\"F&,,*$F(\"\"#F&*&F(F&F)F&!\"#*$F)F-F& *$%\"yGF-F&*$%\"zGF-F&#!\"&F-!\"$*(%\"CGF&,&F(F&F)F&F&,,F,F&F.F-F0F&F1 F&F3F&F5F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "simplify(\", \{x^2-2*x*A+A^2+y^2+z^2=R1^2, x^2+2*x*A+A^2+y^2+z^2=R2^2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&***$%$R1|irG\"\"##!\"&F'*$%$R2|irGF'F(,&*& %\"BG\"\"\"F*#\"\"&F'F/*&%\"CGF/F%F0!\"\"F/%\"AGF/\"\"$**,&F-F/F2F/F/F %F(F*F(%\"xGF/!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Again, this is not what we want. Use " }{TEXT 267 8 "termfunc" }{TEXT -1 1 ":" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "termfunc(\"\",simplify,\{x ^2-2*x*A+A^2+y^2+z^2=R1^2, x^2+2*x*A+A^2+y^2+z^2=R2^2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"BG\"\"\"*$%$R1|irG\"\"##!\"&F),&%\"xG !\"\"%\"AGF&F&\"\"$*(%\"CGF&*$%$R2|irGF)F*,&F-F.F/F.F&F0" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Autbutilitroutinapplyfuncsyntaxexprfuncprocedurfunctnonnegintpurposapplexpresstimeexamplargumentanymaplobjectnumbsincontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsotstringcosfixarracsquarindexequatnumericnamesetdebugprintpnammsgverbositlevelexpansexpressparamorderintegsimplflagfuncopoplistfunctioncallgetsqrtrealpartanythtermpolytestpulloutsubexprrootfuncargsrotaxithetaaxissignsqrtsinfixsmalldivisortermfunctmsgproctopsqrttypefractionalpowercontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoaesopodec_Lom䉬sˬkhGhRh dsPgja+Jf_@dP%@cX)\8 c 9^O@b``fe8ϫ@B^d3@kO-_PMd @sFdˬ&hlfsX@`ʮ@B`pٮ@gj7+J_d" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 260 8 "csquare(" }{TEXT 265 1 " \+ " }{TEXT 263 106 "expr::\{algebraic,array,indexed,equation,numeric\} \+ \n <, var::\{name,set,list,`+`,`*`,`^`,function\}> )" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 158 "Complete the square in the (optional) variable var in the (2nd degree in var) polynomial expr. This is routine is more general than student[completesquare]. " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arg uments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 30 " An exp ression or equation." }}{PARA 15 "" 0 "" {TEXT 258 7 "var " }{TEXT -1 139 "A name, a function, a `+`, `*`, or `^` expression, or a set or \n list of functions, variable names, and `+`, `*`, `^` expr essions." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "e1 := (a*c+b^2-d)*(a*c+b^2+d) = 0: \ne2 := (t*u+p*q+r*s)*(t*u-p*q-r*s):\ne3 := 9*sin(x)^2 + 24*sin(x) + 1 6 = f(x):\ne4 := x^2 - 2*x*a + a^2 + y^2 -2*y*b + b^2 = F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "csquare(e1,b^2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&*$,&*$%\"bG\"\"#\"\"\"*&%\"aGF*%\"cGF*F*F)F**$%\"d GF)!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "csquare(e4 ,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$,&%\"yG\"\"\"%\"bG!\" \"\"\"#F(*$,&%\"xGF(%\"aGF*F+F(%\"FG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "M := array([[e1,e2],[e3,e4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7$7$/*&,(*&%\"aG\"\"\"%\"cGF/F/*$%\"b G\"\"#F/%\"dG!\"\"F/,(F-F/F1F/F4F/F/\"\"!*&,(*&%\"tGF/%\"uGF/F/*&%\"pG F/%\"qGF/F/*&%\"rGF/%\"sGF/F/F/,(F:F/F=F5F@F5F/7$/,(*$-%$sinG6#%\"xGF3 \"\"*FH\"#C\"#;F/-%\"fGFJ/,.*$FKF3F/*&FKF/F.F/!\"#*$F.F3F/*$%\"yGF3F/* &FXF/F2F/FUF1F/%\"FG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "csq uare( M, [p*q,b^2,x,y,sin(x)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'MATRIXG6#7$7$/,&*$,&*$%\"bG\"\"#\"\"\"*&%\"aGF/%\"cGF/F/F.F/*$%\"dGF. !\"\"\"\"!,&*$,&*&%\"pGF/%\"1 "" {XPPMATH 20 "6#,&%\"DG\"\"\"*(%\"AGF%%\"BG\"\"#,&%\"zGF%*&%\"xGF% ,&F%F%*&%\"CGF%%\"yGF%F%F%F%F)F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "A slightly more complicated expression:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "foo := F*A*(B*(1+x)+C*B*y*x+z*B*(x+1))^2+D*x-E*B; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fooG,(*(%\"FG\"\"\"%\"AGF(,(*&% \"BGF(,&F(F(%\"xGF(F(F(**%\"CGF(F,F(%\"yGF(F.F(F(*(%\"zGF(F,F(F-F(F(\" \"#F(*&%\"DGF(F.F(F(*&%\"EGF(F,F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pullout( foo, B*(1+x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"DG\"\"\"%\"xGF&F&*&%\"EGF&%\"BGF&!\"\"*(%\"FGF&% \"AGF&,&**%\"CGF&F*F&%\"yGF&F'F&F&*(F*F&,&F&F&F'F&F&,&F&F&%\"zGF&F&F& \"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "pullout( \", B ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"DG\"\"\"%\"xGF&F&*&%\"BGF& ,&%\"EG!\"\"**%\"FGF&%\"AGF&F)F&,&*(%\"CGF&%\"yGF&F'F&F&*&,&F&F&F'F&F& ,&F&F&%\"zGF&F&F&\"\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\"-foo);" }}{PARA 11 "" 1 "/ individual  indvarchang 7787instead77integYYintegerinto K4YinvolvYisdiff N487kappaYlet77levelYlhs yylinear'list3#Q'O487TTYY [yylocat 87T Ymain makecoeff87mapl# P87TYYbmarcWPQ'K4M4N4O47787TTTTYY[byy7787TTTTYY[byyydebugPYyT ODE,type,diffK4 ODE,varcH;argumentWPQ'K4M4N4O47787TTTTYY[byyarraQ87YY yy artificial77assumY astronomical[PQ'K4M4N4O47787TTTTYY[byyy77oderoutinindvarchangsyntaxdeqnteqnequatnameunknownsetpurpostransformoldindependvariablnewdifferentialmustformsomefunctallowablspecialcasesecondexamplbelowargumentexpresscontainderivatspecifytransformatfunctiontreatsimpreplacrathsolutsimplaccorddiffthetaomegasupposinsteadwewishletunspecifisaydonesettphimorecomplicatartificialgnarchosenspecificalfullexerciscossintanalphacheckbackoriginalsimplifbetacollectnormalfactorsymbolicnonuniquencounterwillpromptbranchsolvchoosfollowsemicoloncontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilit x function addprt{PARAl insert a RealRange in the lattix> is is only needed when another pr8xty in the lattice relates (as parent or child) of a Realist,function\}, order1::inte ger,\n params2::\{string,list,function\}, order2::integer, \n simplflag::integer )" }}{PARA 276 "" 0 "" {HYPERLNK 17 " funcops" 2 "utilities,funcops" "" }{TEXT -1 54 "( expr::`+`, func::pro cedure, oplist::\{integer,list\} )" }}{PARA 271 "" 0 "" {TEXT -1 21 "F unctionCalls( expr )" }}{PARA 262 "" 0 "" {TEXT -1 38 "getsqrts( expr: :\{algebraic,equation\} )" }}{PARA 274 "" 0 "" {TEXT -1 21 "is_real_pa rts( expr )" }}{PARA 286 "" 0 "" {HYPERLNK 17 "location" 2 "utilities, location" "" }{TEXT -1 34 "( expr::anything, term::anything )" }} {PARA 277 "" 0 "" {TEXT -1 51 "polytest( expr::algebraic, varlist::\{l ist,string\} )" }}{PARA 285 "" 0 "" {HYPERLNK 17 "pullout" 2 "utilitie s,pullout" "" }{TEXT -1 66 "( expr::\{`+`,`^`,`*`\}, subexpr::\{name,f unction,`+`,`*`,`^`,list\} )" }}{PARA 269 "" 0 "" {TEXT -1 98 "rootfun c( expr::\{algebraic,equation,list,set,indexed,array\}, \n fu nc::procedure <, args> )" }}{PARA 275 "" 0 "" {TEXT -1 41 "RotAxis( tEPutilitroutintmsgsyntaxprocnamemsgpurposprintmessagformelapstimeterminalscreencurrminustoroptionalthirdargumwithdecimalparttruncatargumentcallprocedurusualprocnamstringmustmaplfiducialexamplsomefunctargsnarghihocontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsodebugpolytestrootfuncvectorargsrotaxithetaaxissignsqrtsmalldivisortmsgproctypefractionalpowercontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoaesopastronomical OPauthor OPave OPaverag OQaxisObefor OQbracket OQcallP combinetrig OQcommon OQcontact OPcoord OQcsquar OQcurrPdc OPda nn.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno .navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" } }{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "termfunc" 2 "utilities,termfunc" "" }{TEXT -1 1 " " }}}} "\"$F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "funcops( expr, factor, [1,3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$,&%\"BG\"\"\"!\"\"F'\"\"#F'*$,&%\"CGF'F'F' \"\"$F'*$%\"AGF)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f uncops( expr, simplify, [1,3], \{A^2+1=D^2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&%\"BG\"\"\"!\"\"F'\"\"#F'*$,&%\"CGF'F'F'\"\"$F'*$ %\"DGF)F'" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author " }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical A pplications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW \nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riem(n " }{TEXT 267 4 "expr" }{TEXT -1 23 " to collectively apply " }{TEXT 268 4 "func" }{TEXT -1 4 " to. " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expr := (A+1)^2 + (B-1)^2 - 2*A + (C+1)^3;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,**$,&%\"AG\"\"\"F)F)\"\"#F)* $,&%\"BGF)!\"\"F)F*F)F(!\"#*$,&%\"CGF)F)F)\"\"$F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "funcops( expr, factor, [1,3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$,&%\"BG\"\"\"!\"\"F'\"\"#F'*$,&%\"CGF'F'F' \"\"$F'*$%\"AGF)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f uncops( expr, simplify, [1,3], \{A^2+1=D^2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&%\"BG\"\"\"!\"\"F'\"\"#F'*$,&%\"CGF'F'F'\"\"$F'*$ %\"DGF)F'" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author " }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical A pplications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW \nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riem(0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 263 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT 265 12 " \226 funcops()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 261 "" 0 "" {TEXT 259 61 "funcops( expr::`+`, fu nc::procedure, oplist::\{integer,list\} )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 131 "Apply a procedure func to individual (top-level) terms in expression expr. the specifi ed operands (as a group) of the expression. " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" } {TEXT -1 33 " Expression to be simplified." }}{PARA 15 "" 0 "" {TEXT 258 8 "func " }{TEXT -1 58 " Maple procedure that will act up on the group of terms of " }{TEXT 262 4 "expr" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT 266 6 "oplist" }{TEXT -1 58 " An integer or a list of integers denoting the terms i) 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE " Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Addre ss" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 *Qiutilitroutincsquarsyntaxexpralgebraicarraindexequatnumericvarnamesetlistfunctpurposcompletsquaroptionalvariablnddegrepolynomialmoregeneralstudcompletesquarargumentexpressfunctionexamplsincontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoproceduroplistfunctioncallgetsqrtrealpartpolytestrootfuncvectorargsrotaxithetaaxissignsqrtsmalldivisortmsgproctypefractionalpowercontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoaesop@sP{SECTJ,(ƞ  HO@|~8O ~`{PARANormalizerJ*X  O>  OrderJ" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Another form results from a differ enct sequence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "pullout( \+ foo, B*x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"DG\"\"\"%\"xGF&F &*&%\"EGF&%\"BGF&!\"\"*,%\"FGF&%\"AGF&F*\"\"#F'F/,(*&,&F&F&F'F&F&F'F+F &*&%\"CGF&%\"yGF&F&*(%\"zGF&F2F&F'F+F&F/F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Oops, still a factor of B to pull out." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pullout(\",B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"DG\"\"\"%\"xGF&F&*&%\"BGF&,&%\"EG!\"\"*,%\"FGF&% \"AGF&F)F&F'\"\"#,(*&,&F&F&F'F&F&F'F,F&*&%\"CGF&%\"yGF&F&*(%\"zGF&F3F& F'F,F&F0F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(\" -foo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval \+ Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }} {PARA 1ableYaccord77act aesop 87Yagain algebraic+(Q'O487TYYyyaliaYallowabl77alpha O477TTYalsoWPQ'K4M4N4O47787TTTTYY[byyalway[angl 87YannularYanoth[answ[any87TYbanyth TYappearYapplbapplicatWPQ'K4M4N4O47787TTTTYY[byy applyfunc Ybargs PYargumPyynwWPQ'K4M4N4;PQ'K4M4N4O47787TTTTYY[byyy;ableYaccord77act aesop87Yyagain algebraic+(Q'O4DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riema nn.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno .navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" } }{PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "utilities" 2 "utilities" "" }}}} bpartPbt2v2[bb)F)*&%\"BGF)F-F)F)%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c := diffcoeffs(ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7$7%%\"AG%\"BG,$%\"CG!\"\"7%-%%diffG6$-F.6$-%\"xG6#%\"tGF5 F5F0\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sum( c[1][k]* c[2][k], k=1..nops(c[1]) ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( *&%\"AG\"\"\"-%%diffG6$-F)6$-%\"xG6#%\"tGF0F0F'F'*&%\"BGF'F+F'F'%\"CG! \"\"\"\"!" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author " }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical A pplications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW \nWashignton, 22 " Expression to \+ be simplified." }}{PARA 15 "" 0 "" {TEXT 258 8 "func " }{TEXT -1 48 "Maple procedure that will act upon the terms of " }{TEXT 266 4 "ex pr" }{TEXT -1 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "r1 := sqrt((x-A)^2+y^2+z^2) : r2 := sqrt((x+A)^2 + y^2 + z^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "expr := diff( B/r1^3 + C/r2^3, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&*(%\"BG\"\"\",,*$%\"xG\"\"#F(*&F+F(%\"AGF( !\"#*$F.F,F(*$%\"yGF,F(*$%\"zGF,F(#!\"&F,,&F+F,F.F/F(#!\"$F,*(%\"CGF(, ,F*F(F-F,F0F(F1F(F3F(F5,&F+F,F.F,F(F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(,* *(%\"BG\"\"\",,*$%\"xG\"\"#F(*&F+F(%\"AGF(F,*$F.F,F(*$%\"yGF,F(*$%\"zG F,F(#\"\"&F,F+F(F(*(F'F(F)F4F.F(!\"\"*(%\"CGF(,,F*F(F-!\"#F/F(F0F(F2F( F4F+F(F(*(F9F(F:F4F.F(F(F(F:#!\"&F,F)F=!\"$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 45 "This is ugly and not what we want, so we use " }{TEXT 268 8 "termfunc2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 13 " \226 termfunc()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 261 "" 0 "" {TEXT 260 10 "termfunc( " }{TEXT 263 47 "expr::\{algebraic,array(1),`=`\}, func::procedure" }{TEXT 261 2 " )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 76 "Apply a procedure func to individual (top-level) terms in expression expr. " }}{PARA 0 "" 0 "" {TEXT -1 83 "Termfunc() is a way to apply a procedure to the main parts of an expression without" }}{PARA 0 "" 0 "" {TEXT -1 57 "messing up the entire expression. See \+ the example below." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 330 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Fon t 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 4-{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 5nargPnavalWPQ'K4M4N4O47787TTTTYY[byynavyW*PQ'K4M4N4O47787TTTTYY[byyndQnew77no Y[nonintegY nonnegint Ybnonuniqu77nops'normal77note87YY[notic[numbbnumeric QYnwWPQ'K4M4N4O47787TTTTYY[byyyugly univariatO4unknown 7787 unspecifi77up ;;PQ'K4M4N4O47787TTTTYY[byyy; differenct[ differential O477diffexpr M487difford M487diffvar87TTdivisorYdn'doesYdone77YySutilitroutincosfixsyntaxexpralgebraicarravarlistpurposforcargumentexpresssimplifiargumcosinfunctionsimplifexamplrhotaudiffthetaphicosfactortermfunclhsrhscollectcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsosinfix massachusettWPQ'K4M4N4O47787TTTTYY[byymatrix YY maxpdeord 87TmessmessagPmessiO4milW*PQ'K4M4N4O47787TTTTYY[byyminuPmoreQ77[msg PYmultiplYmurisonW?PQ'K4M4N4O47787TTTTYY[byymustP7787YnamePQ7787YY[T ODE,type,diffK4 ODE,varchangeO4 utilitiesyutilities,applyfunc`;func;PQ'K4M4N4O47787TTTTYY[byyy4 ; individual  indvarchang 7787instead77integYYintegerinto K4YinvolvYisdiff N487kappaYToderoutindepvarsyntaxexprdiffpurposreturndependvariablderivatargumentexamplcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitdiffvarT&oderoutinmaxpdeordsyntaxprocexpralgebraicpurposreturnorderhighestderivatexpressequatargumentcontainexamplcosdifftanalphasincontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitEyutilitroutintheslocatfileutilmustreadbeforusinganyexamplmaplsyntaxnoteargumentenclosanglbracketoptionalannularaveragexprcoordparamlistapplyfuncfuncprocedurfunctnonnegintcombinetrigalgebraicvaroptimizcombinetrigoptaliacommonfactorvarlistliststringcosfixarracsquarindexequatnumericnamesetdebugprintpnammsgverbositlevelexpansexpressparamorderintegsimplflagfuncopoplistfunctioncallgetsqrtrealpartanythtermpolytestpulloutsubexprrootfuncargsrotaxithetaaxissignsqrtsinfixsmalldivisortermfunctmsgproctopsqrttypefractionalpowercontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoaesopodeV\VGlVVGVxVGVV C )" }}{PARA 282 "" 0 "" {TEXT -1 38 "topsqrts( ex pr::\{algebraic,equation\} )" }}{PARA 281 "" 0 "" {TEXT -1 30 "topsqrt ( expr::\{`+`,`*`,`^`\} )" }}{PARA 268 "" 0 "" {TEXT -1 76 "`type/frac tional_power`( expr::\{algebraic,equation,list,set,indexed,array\} )" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "D<decimalPdegreQdenotdependO4TTdeptWPQ'K4M4N4O47787TTTTYY[byydepvar87TTdeqn O47787derivat#'M4N4O477TTTdescribO4determin M4N4dexO4deyO4dezO4diff;G'K4 M4N4O4 7787TTTTyy diffcoeff '87diffeqn8787TTTTYY[byyyn 7787termTYYytermfunc Y[yyyterminalPthes 87Ytheta,O477YYyyythirdPtime PbtmsgPYytop TTY[byyyndQnew77no Y[nonintegY nonnegintY ;can'ERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "AESOP" 2 "AESOP" "" }{TEXT -1 2 " " }}}} @<m0TODE,diffcoeffs' ODE,diffo269 "" 0 "" {HYPERLNK 17 "maxPDEorde r" 2 "ODE,maxPDEorder" "" }{TEXT -1 25 "( expr::\{algebraic,`=`\} )" } }{PARA 264 "" 0 "" {TEXT -1 19 "Reconstruct( expr )" }}{PARA 270 "" 0 "" {HYPERLNK 17 "`type/diff`" 2 "ODE,type,diff" "" }{TEXT -1 5 "( x ) " }}{PARA 273 "" 0 "" {HYPERLNK 17 "varchange" 2 "ODE,varchange" "" } {TEXT -1 58 "( deqns::\{algebraic,`=`,list,set\}, trans::\{`=`,set,lis t\} )" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }} {PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appli cations Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWa shignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.u sno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.nav y.mil/murison/" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPG 263 "" 0 " " {HYPERLNK 17 "isdiff" 2 "ODE,isdiff" "" }{TEXT -1 5 "( x )" }}{PARA 268 "" 0 "" {TEXT -1 67 "MakeCoeffs( expr::\{algebraic,algebraic=algeb raic,list,'array'(1)\} )" }}{PARA 269 "" 0 "" {HYPERLNK 17 "maxPDEorde r" 2 "ODE,maxPDEorder" "" }{TEXT -1 25 "( expr::\{algebraic,`=`\} )" } }{PARA 264 "" 0 "" {TEXT -1 19 "Reconstruct( expr )" }}{PARA 270 "" 0 "" {HYPERLNK 17 "`type/diff`" 2 "ODE,type,diff" "" }{TEXT -1 5 "( x ) " }}{PARA 273 "" 0 "" {HYPERLNK 17 "varchange" 2 "ODE,varchange" "" } {TEXT -1 58 "( deqns::\{algebraic,`=`,list,set\}, trans::\{`=`,set,lis t\} )" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }} {PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appli cations Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWa shignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.u sno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.nav y.mil/murison/" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPG"" {TEXT 257 12 "ODE Routines" }}{PARA 0 "" 0 "" {TEXT -1 125 "These routines are located in the file diffeqn.p. This f ile must be read in before using any of these routines. For example, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "read`c:/maple/diffeqn.p` ;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 20 60 "NOTE: Arguments enclosed in angle brackets <> are option al." }}{PARA 265 "" 0 "" {HYPERLNK 17 "diffcoeffs" 2 "ODE,diffcoeffs" "" }{TEXT -1 41 "( expr::\{algebraic,algebraic=algebraic\} )" }}{PARA 266 "" 0 "" {HYPERLNK 17 "difforder" 2 "ODE,difforder" "" }{TEXT -1 18 "( diffexpr::diff )" }}{PARA 271 "" 0 "" {HYPERLNK 17 "indvarchange " 2 "ODE,indvarchange" "" }{TEXT -1 2 "( " }{TEXT 258 56 "deqn, Teqn:: equation, t::name, T::name <, unknowns::set>" }{TEXT -1 2 " )" }} {PARA 272 "" 0 "" {HYPERLNK 17 "depvar" 2 "ODE,depvar" "" }{TEXT -1 18 "( diffexpr::diff )" }}{PARA 267 "" 0 "" {HYPERLNK 17 "diffvars" 2 "ODE,diffvars" "" }{TEXT -1 18 "( diffexpr::diff )" }}{PARAHLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 266 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 267 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 268 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 269 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 270 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 271 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 272 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 273 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 IMPLTEXT 1 0 20 "f := (x,a ) -> a*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"aG6\"6 $%)operatorG%&arrowGF)*&9%\"\"\"9$\"\"#F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "applyfunc( B*y+C*z, f, 2, A );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%\"AG\"\"$,&*&%\"BG\"\"\"%\"yGF)F)*&%\"CGF)%\"zGF)F )\"\"%" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" } }{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appl ications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nW ashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann. usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.na vy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 1 " " }}}} RA 15 "" 0 "" {TEXT 258 4 "func" }{TEXT -1 34 " A MapPARA 0 "> " 0 "" {MPLTEXT 1 0 23 "applyfunc( x, sin, 3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#- F$6#-F$6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {K1 261 1 {CSTYLE "" -1 -1 "Co urier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Add ress - email" 259 263 1 {CSTYLE "" -1 -1 "Verdana" 1 11 0 0 128 1 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address" 0 264 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 265 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 266 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 267 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 268 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 269 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 YAutilitroutinexpanssyntaxexprlistsetarramatrixalgebraicparamnamefunctorderintegpurposexpandintoseriupincludtermsmallparameternoperformmultiplindependcandonestepexamplthetaepsilonpsialphaphibetanotemaplbugsuchsometimcharactertypenonintegpowerinvolvexpresshavenablefiguroutworkaroundhavetriedconvertfailappearconverspolynomdoesremovargumentwillfoosqrtkappacossinassumcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalso`> PJJ J4JpJOJJhTPJJPJTTTPJlYY  J4JOJpJJ`>PJhJOTPJOPJTUU8UPYT  0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 262 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 11 " \226 cosfix()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 263 "" 0 "" {TEXT 259 8 "cosfix( " }{TEXT 261 53 "e xpr::\{algebraic,array(1),`=`\}, var::\{algebraic,list\}" }{TEXT 260 2 " )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 6 "Force " }{XPPEDIT 18 0 "1-cos^2" ",&\"\"\"\"\"\"*$%$cos G\"\"#!\"\"" }{TEXT -1 5 " ==> " }{XPPEDIT 18 0 "sin^2" "*$%$sinG\"\"# " }{TEXT -1 2 ". " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" } }{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 32 " Expression to b e simplified." }}{PARA 15 "" 0 "" {TEXT 258 9 "var " }{TEXT -1 49 "The argument of the cosine functions to simplify." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "-rho(tau)*diff(theta(tau),tau)^2-rho(tau)*diff(phi(t au),tau)^2-2*rho(tau)*diff(phi(tau),tau)+rho(tau)*c\ 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 3 " \226 " }{TEXT 261 9 "applyfunc" }{TEXT -1 2 "()" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 260 59 "applyfunc( expr, func::\{procedure,function\}, n::nonnegint ) " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 61 "Apply a function to an expression n times (see the Exampl es)." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 35 " Any expression or Maple obje ct." }}{PARA 15 "" 0 "" {TEXT 258 4 "func" }{TEXT -1 34 " A Maple f unction or procedure." }}{PARA 15 "" 0 "" {TEXT 259 2 "n " }{TEXT -1 37 " The number of times to apply " }{TEXT 262 4 "func" }{TEXT -1 4 " to " }{TEXT 263 4 "expr" }{TEXT -1 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "applyfunc( x, sin, 3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#- F$6#-F$6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {K 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 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-1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4PE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Co urier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 3 " \226 " }{TEXT 277 11 "maxPDEorder" }{TEXT -1 2 "()" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 258 44 "maxPDEokcan'Y[case 77[ characterYcheck 77[choos77chosen77cn' coefficient'collect77yy collective combinetrigYcombinetrigoptYcommonYcompletQ completesquarQ complicat 77[contactWPQ'K4M4N4O47787TTTTYY[byycontain'O477TconversYconvertYcoordY correspond'cos O477TTYycosfixYyycosin yycours[csquar QYcurrPdcWPQ'K4M4N4O47787TTTTYY[byydebug PYY polynomialQpolytestYpossiblTpowerYYyprint PY87oderoutintheslocatfilediffeqnmustreadbeforusinganyexamplmaplsyntaxnoteargumentenclosanglbracketoptionaldiffcoeffexpralgebraicdifforddiffexprdiffindvarchangdeqnteqnequatnameunknownsetdepvardiffvarisdiffmakecoefflistarramaxpdeordreconstructtypevarchangtrancontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitaesoppolyJ{ ǃgetuserinterfaceJ   ǃgotoJlyH ǃhas IBM INTEL NT \_floatJ"-` Maple Library Help ODE,indvarchange \_indexedJXPPfunction ist@ wP*_`*_``al  IBM INTEL NT77Tinstead77integ ( PO`aP)2`3`@@` 0 "" {TEXT -1 9 "A rguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 53 " An e xpression or equation containing derivatives." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode := A*diff(x(t),t,t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$odeG/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2 F2F)F)*&%\"BGF)F-F)F)%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c := diffcoeffs(ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7$7%%\"AG%\"BG,$%\"CG!\"\"7%-%%diffG6$-F.6$-%\"xG6#%\"tGF5 F5F0\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sum( c[1][k]* c[2][k], k=1..nops(c[1]) ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( *&%\"AG\"\"\"-%%diffG6$-F)6$-%\"xG6#%\"tGF0F0F'F'*&%\"BGF'F+F'F'%\"CG! \"\"\"\"!" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author " }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical A pplications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW \nWashignton, 2 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 15 " \226 diffcoeffs()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 258 51 "diffcoeffs( expr::\{algebraic ,algebraic=algebraic\} )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpo se" }}{PARA 0 "" 0 "" {TEXT -1 61 "Return the coefficients of the deri vatives in the expression " }{TEXT 259 4 "expr" }{TEXT -1 94 ", which \+ can be an algebraic expression or an equation. The returned structure is of the form " }}{PARA 0 "" 0 "" {TEXT 260 42 " L = [ [c1,c2,... ,cn], [d1,d2,...,dn] ]" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 261 4 "L[1]" }{TEXT -1 33 " is the list of coefficients and " }{TEXT 262 4 "L[2]" }{TEXT -1 64 " is the list of corresponding derivatives. \+ Hence, the original " }{TEXT 264 6 "linear" }{TEXT -1 27 " ODE can be recovered with " }}{PARA 0 "" 0 "" {TEXT 263 48 " sum( L[1][k]*L[2 ][k], k=1..nops(L[1]) ) = 0;" }}}{SECT 0 {PARA 5 ""V0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 " " 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Add ress" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0W'^{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 Xson@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilitie s" 2 "utilities" "" }}}} @ @`lܷ{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "isdiff( diff(x(t),t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode := A*diff(x(t),t, t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ode G/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2F2F)F)*&%\"BGF)F-F)F)% \"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "isdiff(od e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval \+ Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }} {PARA 15 "" 0 "" {TEXT 256 29 "muriZu)^2+y(tau)^2+z(tau)^2)^(3/2))*y(tau);\ndez0 := diff( diff(z(tau),tau),tau)+z(tau) = (1-1/(x(tau)^2+y(tau)^2+z(tau)^2)^(3/2) )*z(tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dex0G/,&-%%diffG6$-F(6 $-%\"xG6#%$tauGF/F/\"\"\"-F(6$-%\"yGF.F/!\"#*&,&F0F0*$,(*$F,\"\"#F0*$F 3F;F0*$-%\"zGF.F;F0#!\"$F;!\"\"F0F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dey0G/,&-%%diffG6$-F(6$-%\"yG6#%$tauGF/F/\"\"\"-F(6$-%\"xGF.F/\" \"#*&,&F0F0*$,(*$F3F5F0*$F,F5F0*$-%\"zGF.F5F0#!\"$F5!\"\"F0F,F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dez0G/,&-%%diffG6$-F(6$-%\"zG6#%$ta uGF/F/\"\"\"F,F0*&,&F0F0*$,(*$-%\"xGF.\"\"#F0*$-%\"yGF.F8F0*$F,F8F0#! \"$F8!\"\"F0F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "subsli st := \{ x(tau)=rho(tau)*sin(theta(tau))*cos(phi(tau)),\n \+ y(tau)=rho(tau)*sin(theta(tau))*sin(phi(tau)),\n z(tau)= rho(tau)*cos(theta(tau)) \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "varchange( [dex0,dey0,dez0], subslist );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/,&-%%diffG6$-F'6$*(-%$rhoG6#%$txos(theta(tau))^2+di ff(diff(rho(tau),tau),tau)+rho(tau)*diff(phi(tau),tau)^2*cos(theta(tau ))^2+2*rho(tau)*diff(phi(tau),tau)*cos(theta(tau))^2 = (rho(tau)^3-1)/ rho(tau)^2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&-%$rhoG6#%$tauG\" \"\"-%%diffG6$-%&thetaGF(F)\"\"#!\"\"*&F&F*-F,6$-%$phiGF(F)F0F1*&F&F*F 3F*!\"#*&F&F*-%$cosG6#F.F0F*-F,6$-F,6$F&F)F)F**(F&F*F3F0F:F0F**(F&F*F3 F*F:F0F0*&,&*$F&\"\"$F*F1F*F*F&F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "cosfix( \", theta(tau), factor );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,**&-%$rhoG6#%$tauG\"\"\"-%%diffG6$-%&thetaGF(F)\"\" #!\"\"-F,6$-F,6$F&F)F)F*F&F**&,(F&F1*&F&F*-F,6$-%$phiGF(F)F0F1*&F&F*F9 F*!\"#F*-%$sinG6#F.F0F**&,&*$F&\"\"$F*F1F*F*F&F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "termfunc( lhs(\"), factor ) = rhs(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&-%$rhoG6#%$tauG\"\"\"-%%diffG6$-% &thetaGF(F)\"\"#!\"\"-F,6$-F,6$F&F)F)F*F&F**(F&F*,&-F,6$-%$phiGF(F)F*F *F*F0-%$sinG6#F.F0F1*&,&*$F&\"\"$F*F1F*F*F&!\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT ne the order of a derivative." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 258 4 "diff" } {TEXT -1 17 " A derivative." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "difforder( diff (x(t),t,t,t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "difforder( diff(y(t,u),t,u,t,t) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr . Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval Observa tory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }}{PARA 15 " " 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 " " {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 " ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilities" 2 "utilitie s" "" }}}}  di P]T -1 36 "Determine the order of a derivative." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 258 4 "diff" } {TEXT -1 17 " A derivative." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "difforder( diff (x(t),t,t,t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "difforder( diff(y(t,u),t,u,t,t) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr . Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval Observa tory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }}{PARA 15 " " 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 " " {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 " ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilities" 2 "utilitie s" "" }}}}  di P] 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "D ate" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 14 " \226 difforder()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 " Syntax" }}{PARA 0 "" 0 "" {TEXT 257 27 "difforder( diffexpr::diff )" } }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEX^M4 {VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0_'`oderoutindiffcoeffsyntaxexpralgebraicpurposreturncoefficientderivatexpresscanequatstructurformcndnlistcorrespondhencoriginallinearrecoverwithsumnopsargumentcontainexampldiffcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitؙ`ڶ۶@ܶܶTyuĈH f an expression is a derivative." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 1 "x" }{TEXT -1 18 " An \+ expression." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "isdiff( A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "isdiff( diff(x(t),t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode := A*diff(x(t),t, t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ode G/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2F2F)F)*&%\"BGF)F-F)F)% \"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "isdiff(od e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval \+ Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }} {PARA 15 "" 0 "" {TEXT 256 29 "muriZ 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "D ate" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 11 " \226 isdiff()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syn tax" }}{PARA 0 "" 0 "" {TEXT 258 11 "isdiff( x )" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 43 "Determine icN4$ {VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0dK4 {VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0g 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "D ate" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 17 " \226 `type/diff `()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 258 16 "`type/diff`( x )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 33 "h Enter \"diff\" into the type table." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 1 "x" }{TEXT -1 18 " \+ An expression." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "type( A, diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "type( diff(x(t),t), diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "o de := A*diff(x(t),t,t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2F2 F)F)*&%\"BGF)F-F)F)%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "type( ode, diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "has( ode, diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applicatioins Dept.\nU.S. Naval \+ Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }} {PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilitie s" 2 "utilities" "" }}}} @@HXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "o de := A*diff(x(t),t,t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2F2 F)F)*&%\"BGF)F-F)F)%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "type( ode, diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "has( ode, diff );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applicatioi -1 123 "Dr. Marc A. Murison \nAstronomical Applications Dept.\nU.S. Naval Observatory\n3450 Massac husetts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "ht tp://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "utilities" 2 "utilities" "" }}}} 4T + \n t*(A(t)-B(t))*tan(t)*(G(t)-diff(F(t),t,t,t))^2 *diff(x(t),t) \n = alpha*t*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&*(-%\"AG6#%\"tG\"\"\"-%$cosGF*F,-%%diffG6$-F06$-%\"xGF *F+F+F,F,*,F+F,,&F(F,-%\"BGF*!\"\"F,-%$tanGF*F,,&-%\"GGF*F,-F06$-F06$- F06$-%\"FGF*F+F+F+F:\"\"#F2F,F,*(%&alphaGF,F+F,-%$sinGF*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "maxPDEorder( ode );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Co ntact the Author" }}{PARA 15 "" 0 "" {TEXTjrder := proc( expr::\{algebraic,`=`\} )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 78 "Re turn the order of the highest-order derivative in an expression or equ ation." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 " " 0 "" {TEXT 257 4 "expr" }{TEXT -1 53 " An expression or equation \+ containing derivatives." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Exampl es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "ode := A(t)*cos(t)*di ff(x(t),t,t) + \n t*(A(t)-B(t))*tan(t)*(G(t)-diff(F(t),t,t,t))^2 *diff(x(t),t) \n = alpha*t*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&*(-%\"AG6#%\"tG\"\"\"-%$cosGF*F,-%%diffG6$-F06$-%\"xGF *F+F+F,F,*,F+F,,&F(F,-%\"BGF*!\"\"F,-%$tanGF*F,,&-%\"GGF*F,-F06$-F06$- F06$-%\"FGF*F+F+F+F:\"\"#F2F,F,*(%&alphaGF,F+F,-%$sinGF*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "maxPDEorder( ode );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Co ntact the Author" }}{PARA 15 "" 0 "" {TEXTjEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "op()" 2 "op" "" }{TEXT -1 1 " " }}}} <Od0 "6#7'\"\"\"F$\"\"$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "loca tion( ode, (diff(x(t),t))^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"\"\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%diffG6$-%\"xG6#%\"tGF*\"\"# " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }} {PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appli cations Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWa shignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.u sno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.nav y.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {Tl"\"\"\"%\"\"&F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"FG6#%\"tG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "location( ode, x(t) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"F$\"\"$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "loca tion( ode, (diff(x(t),t))^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"\"\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%diffG6$-%\"xG6#%\"tGF*\"\"# " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }} {PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appli cations Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWa shignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.u sno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.nav y.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {Tlsay77screenPsecond77 selective[ semicolon77sensTsequenc[seriYsetQO47787YYsett77 significant[signsqrtYsimp77simpl O477simplif O477yysimplifiyysin#QO477TTYbysinfixYyyslight[small YYsolut77solv77some77 somefunctPsometimYspecial77specifi specifical77specify77sqrt YsquarQstack[step Y[YbargsPYyargumPyy87Ybelow77[beta 77YbitO4bracket87Yybranch77bugYbutstill[storPstring PYstructur'studQsubexpr Y[ subexpress[subsO4subslistO4suchYsuitablTsum'suppos77symbolic O477syntaxWPQ'K4M4N4O47787TTTTYY[byytablK4tan77TTtauYO4%yyteqn 7787termTYYtermfunc Y[yyterminalPthes 87Ytheta+O477YYyythirdPtime Pbtmsg PYtop ycurrPdc[PQ'K4M4N4O47787TTTTYY[byyydebugPYyut[callP ;argumentWPQ'K4M4N4O47787TTTTYY[byyarraQ87YY yy artificial77assumY 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>PQ'K4M4N4O47787TTTTYY[byyyK4oderoutintypediffsyntaxpurposenterintotablargumentexpressexamplcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitM4oderoutindiffordsyntaxdiffexprdiffpurposdeterminorderderivatargumentexamplcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitN4oderoutinisdiffsyntaxpurposdeterminexpressderivatargumentexampldiffcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitO4oderoutinvarchangsyntaxdeqnalgebraiclistsettranpurpostransformdependvariablunivariatdifferentialequatargumentexpressequationcontainderivatdescribtransformatexamplsimplcosdiffsinalphavalubitmessidextaudeydezsubslistrhothetaphisubssimplifsymboliccontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilit*cos(t)*diff (x(t),t) - C = alpha*t*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$o deG/,(*(%\"AG\"\"\"-%$cosG6#%\"tGF)-%%diffG6$-F/6$-%\"xGF,F-F-F)F)**% \"BGF)-%$sinGF,F)F*F)F1F)F)%\"CG!\"\"*(%&alphaGF)F-F)F7F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "varchange(ode,x(t)=u(t)-v(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"AG\"\"\"-%$cosG6#%\"tGF'-%%dif fG6$-F-6$,&-%\"uGF*F'-%\"vGF*!\"\"F+F+F'F'**%\"BGF'-%$sinGF*F'F(F'F/F' F'%\"CGF6*(%&alphaGF'F+F'F9F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"AG\"\"\"-%$ cosG6#%\"tGF',&-%%diffG6$-F.6$-%\"uGF*F+F+F'-F.6$-F.6$-%\"vGF*F+F+!\" \"F'F'**%\"BGF'-%$sinGF*F'F(F',&F0F'F6F:F'F'%\"CGF:*(%&alphaGF'F+F'F=F '" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "\nAn example that is a bit m essier." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "dex0 := diff(di ff(x(tau),tau),tau)-2*diff(y(tau),tau) = (1-1/(x(tau)^2+y(tau)^2+z(tau )^2)^(3/2))*x(tau);\ndey0 := diff(diff(y(tau),tau),tau)+2*diff(x(tau), tau) = (1-1/(x(ta[260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 14 " \226 varchange()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 " Syntax" }}{PARA 0 "" 0 "" {TEXT 258 67 "varchange( deqns::\{algebraic, `=`,list,set\}, trans::\{`=`,set,list\} )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 39 "Transform the depe ndent variables of a " }{TEXT 259 10 "univariate" }{TEXT -1 23 " diffe rential equation." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" } }{PARA 15 "" 0 "" {TEXT 257 5 "deqns" }{TEXT -1 80 " An expression, equation, or list or set of equations containing derivatives." }} {PARA 15 "" 0 "" {TEXT 260 5 "trans" }{TEXT -1 113 " An equation o r set or list of equations that describe the transformation(s) of the \+ dependent variable(s) in " }{TEXT 261 5 "deqns" }{TEXT -1 1 "." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 17 "A simple example." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "ode := A*cos(t)*diff(x(t),t,t) + B*sin(t)s" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA tO4{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "No rmal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 } 0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 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0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLSauG\"\"\"-%$sinG6#-%& thetaGF.F0-%$cosG6#-%$phiGF.F0F/F/F0-F'6$*(F,F0F1F0-F2F8F0F/!\"#**,&F0 F0*$,(*(F,\"\"#F1FEF6FEF0*(F,FEF1FEF>FEF0*&F,FE-F7F3FEF0#!\"$FE!\"\"F0 F,F0F1F0F6F0/,&-F'6$F;F/F0F)FE**FAF0F,F0F1F0F>F0/,&-F'6$-F'6$*&F,F0FHF 0F/F/F0FWF0*(FAF0F,F0FHF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs( %1=simplify(%1,symbolic), \" );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/,8*(-%%diffG6$-F(6$-%$rhoG6#%$tauGF/F/\"\"\"-%$sinG6 #-%&thetaGF.F0-%$cosG6#-%$phiGF.F0F0**F*F0-F7F3F0-F(6$F4F/F0F6F0\"\"#* *F*F0F1F0-F2F8F0-F(6$F9F/F0!\"#**F,F0F1F0F=F?F6F0!\"\"**F,F0F " 0 "" {MPLTEXT 1 0 23 "diff(F(t,y,z),t, y,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-% \"FG6%%\"tG%\"yG%\"zGF/F0F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "depvar(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"FG6%%\"tG%\"y G%\"zG" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" } }{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appl ications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nW ashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann. usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.na vy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "utilities" 2 "utilities"} "" }{TEXT -1 2 ", " }{HYPERLNK 17 "diffvars" 2 "ODE,diffvars" "" }{TEXT -1 2 " " }}}}  } {TEXT -1 17 " A derivative." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(F(t,y,z),t, y,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-% \"FG6%%\"tG%\"yG%\"zGF/F0F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "depvar(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"FG6%%\"tG%\"y G%\"zG" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" } }{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Appl ications Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nW ashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann. usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.na vy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }} {PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "utilities" 2 "utilities"}ARA 15 "" 0 "" {TEXT 258 8 "term " }{TEXT -1 17 "The term (in the " }{HYPERLNK 17 "op()" 2 "op" "" }{TEXT -1 25 " sense) to locate within " }{TEXT 267 4 "expr" }{TEXT -1 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Exampl es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "ode := A(t)*cos(t)*di ff(x(t),t,t) + \n B(t)*sin(t)*cos(t)*diff(x(t),t)^2 - C(t) - \n \+ t*(A(t)-B(t))*tan(t)*diff(F(t),t)^2*diff(x(t),t) = alpha*t*sin(t );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$odeG/,**(-%\"AG6#%\"tG\"\"\"- %$cosGF*F,-%%diffG6$-F06$-%\"xGF*F+F+F,F,**-%\"BGF*F,-%$sinGF*F,F-F,F2 \"\"#F,-%\"CGF*!\"\"*,F+F,,&F(F,F7F>F,-%$tanGF*F,-F06$-%\"FGF*F+F;F2F, F>*(%&alphaGF,F+F,F9F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "l ocation( ode, B(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"# F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(\",ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"BG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "location( ode, F(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\mTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 13 " \226 location()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 260 42 "location( expr::anything , term::anything )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }} {PARA 0 "" 0 "" {TEXT -1 26 "Find the op() location of " }{TEXT 261 4 "term" }{TEXT -1 4 " in " }{TEXT 262 4 "expr" }{TEXT -1 3 ". " } {TEXT 263 8 "location" }{TEXT -1 84 " returns a Maple list\nsuitable f or use in op(). Hence, if it is possible to return " }{TEXT 266 4 "te rm" }{TEXT -1 47 " via \nthe op() function, then we have that\n " } {TEXT 265 38 "op( location(expr,term), expr ) = term" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 25 " Any Maple expression." }}{P~{CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 259 1 {CSTGutilitroutinlocatsyntaxexpranythtermpurposfindopreturnmapllistsuitablusehencpossiblviafunctwehaveargumentanyexpresssenswithinexamplodecosdiffsintanalphacontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoT{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "No rmal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 } 0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 YLE "" 15 288 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 257 16 "Utility Routines" }}{PARA 0 "" 0 "" {TEXT -1 131 "These utility routines are located in the file util s.p. This file must be read in before using any of these routines. Fo r example," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "read`c:/maple/ utils.p`;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 " " 0 "" {TEXT 20 60 "NOTE: Arguments enclosed in angle brackets <> are optional." }}{PARA 273 "" 0 "" {TEXT -1 48 "annular_average( expr, co ords, a, b, paramlist )" }}{PARA 270 "" 0 "" {HYPERLNK 17 "applyfunc" 2 "utilities,applyfunc" "" }{TEXT 266 50 "( expr, func::\{procedure,fu nction\}, n::nonnegint )" }}{PARA 270 "" 0 "" {TEXT -1 70 "CombineTrig ( expr::\{algebraic,`=`\}, var1::algebraic, var2::algebraic )" }} {PARA 272 "" 0 "" {TEXT -1 30 "`CombineTrig/optimize`( expr )" }} {PARA 283 "" 0 "" {TEXT -1 24 "CombineTrigOpt( expr ) " }{TEXT 2 "" 15 279 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 280 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 281 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 282 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 283 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 284 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 285 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 286 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 287 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }" 15 270 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 271 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 272 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 273 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 274 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 275 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 276 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 277 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 278 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSYLE "Address" 0 261 1 {CSTYLE " " -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 262 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 263 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 264 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 265 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 266 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 267 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 268 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 269 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTY1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Add ress" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 128 1 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE " " -1 -1 "" 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" 256 260 1 {CSTYLE "" -1 -1 "Verdana" 1 11 0 0 128 1 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }y&{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 0 0 0 128 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 128 0 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courie r" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Co urier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -YY{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 128 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 equationO4examplW PQ'K4M4N4O47787TTTTYY[byyexecut[exercis77expandYexpans Y Yexpr?MQ'87TTTTYY[byyexpressCQ'K4N4O477TTYY[byyfactor77Y[ yyfailYfiducialPfigurYfile 87YfindTfollow77foo Y[ forc yyformP'77T[ fractionalYfull77T[ fractional Yyfull77TYY[byyy applyfuncYbyargsPYyargumPyybelow77[beta 77YbitO4bracket87Yybranch77bugYbut[callx ;{CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 128 1 1 2 0 0 0 0 0 0 0 }3 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 8 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" -1 258 1 {CSTYLE "" -1 -1 "V erdana" 1 10 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address" 0 259 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 259 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 261 1 {CSTYLE "" -1 -1 "Co urier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 263 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 14 " \226 expansion()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 259 155 "expansion( expr::\{list ,set,'array'(1),'array'(2),matrix,`=`,algebraic\},\n params: :\{set,name,list,function\}, \n expansion_order::integer )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 7 "Expand " }{TEXT 266 4 "expr" }{TEXT -1 50 " into a series \+ up to and including terms of order " }{TEXT 267 15 "expansion_order" } {TEXT -1 25 " in the small parameters " }{TEXT 268 6 "params" }{TEXT -1 5 ". If " }{TEXT 269 15 "expansion_order" }{TEXT -1 534 " < 1, then no expansions are performed.\n\nMultiple independent expansions can b e done in one step. For example,\n\n expansion( expr, theta, 3, [e psilon,psi,alpha], 2, [phi,beta], 5 );\n\nNOTE: there is a Maple bug s uch that sometimes s is not characterized as a \"series\" type if ther e are noninteger powers of params involved in expression. I haven't be en able to figure out a workaround for this. I have tried convert(s,s eries,params), which fails. When the bug appears, conversion of s to \+ type polynom does not remove the O() term." }}}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 50 " The expression to be expanded in a series." }}{PARA 15 "" 0 "" {TEXT 258 9 "params " }{TEXT -1 61 "A name or a list of name s that are the small parameters that " }{TEXT 270 4 "expr" }{TEXT -1 21 " will be expanded on." }}{PARA 15 "" 0 "" {TEXT 271 15 "expansion_ order" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "foo := 1/sqrt(epsilon*x^2+kappa*y^2+alpha *p^2*cos(theta)-beta*q^3*sin(phi)+A^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fooG*$,,*&%(epsilonG\"\"\"%\"xG\"\"#F)*&%&kappaGF)%\"yGF+F)*( %&alphaGF)%\"pGF+-%$cosG6#%&thetaGF)F)*(%%betaGF)%\"qG\"\"$-%$sinG6#%$ phiGF)!\"\"*$%#A|irGF+F)#F>F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(A>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "expan sion( foo, theta, 4, [epsilon,kappa,alpha,beta], 2, phi, 5 );" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&,**,%#A|irG!\"&%%betaG\"\"\"%\"qG \"\"$%\"xG\"\"#%(epsilonGF*#!\"\"\"$g\"*(,(*(F'F(%\"pGF.F+F,F0**F'F(F6 F.F+F,%&thetaGF.#F*\"$?$**F'F(F6F.F+F,F8\"\"%#F1\"%SQF*F)F*%&alphaGF*F **,F'F(F)F*F+F,%\"yGF.%&kappaGF*F0*(F'!\"$F)F*F+F,#F*\"$S#F*%$phiG\"\" &F***F'F(F)F.F+\"\"'FGF<#F1\"topsqrtYtran O487 transform O477 transformat O477treat77triedYtruncatPtrying[type K487YYugly univariatO4unknown 7787 unspecifi77up Yupon use Tusing 87YusnoW*PQ'K4M4N4O47787TTTTYY[byyusualPutilYutilitW PQ'K4M4N4O47787TTTTYY[byy univariatO4unknown 7787 2PQ'K4M4N4O47787TTTTYY[byyyY;argumentWPQ'K4M4N4O47787TTTTYY[byyarraQ87YY yy artificial77assumY astronomical[PQvaluO4varQTYyyvarchang O487variablQO477 TTvarlistYvarnTverbositYviaTwant washigntonWPQ'K4M4N4O47787TTTTYY[byywaywe77T[will77Ywish77with P'withinTwithout workaroundYOPQ'K4M4N4O477TTTTY[byyO4 7787TTTTyy diffcoeff '87diffeqn87'K4M4N4O47787TTTTY[byyy>;topsqrt Yytr;ableP;Yy ;equationO4examplW PQ'K4M4N4O47787TTTTYY[byyexecut[1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Fon t 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 13 " \226 diffvars()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "S yntax" }}{PARA 0 "" 0 "" {TEXT 258 22 "diffvars( expr::diff )" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 81 "Return the variables of a derivative. The return quantity is a list of the form " }{TEXT 259 20 "[var0,var1,...,varn]" }{TEXT -1 7 " where " }{TEXT 260 4 "var0" }{TEXT -1 31 " is the dependent variable \+ and " }{TEXT 261 10 "var1..varn" }{TEXT -1 58 " are the n independent \+ variables, where n is the order of " }{TEXT 262 4 "expr" }{TEXT -1 23 ". Hence, for example, " }{TEXT 263 22 "diff( y(x,t), x$2, t )" } {TEXT -1 15 " would produce " }{TEXT 264 19 "z := [y(x,t),x,x,t]" } {TEXT -1 8 ". Then " }{TEXT 265 13 "diff( op(z) )" }{TEXT -1 39 " wou ld recover the original derivative." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 17 " \+ A derivative." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Example" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(F(t,y,z),t,y,y,z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-%\"FG6%%\"tG %\"yG%\"zGF/F0F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diffv ars(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'-%\"FG6%%\"tG%\"yG%\"zGF )F(F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff( op(\") ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-%\"FG6%% \"tG%\"yG%\"zGF/F0F0F1" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Conta ct the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nA stronomical Applications Dept.\nU.S. Naval Observatory\n3450 Massachus etts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 " murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http:/ /riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "depvar" 2 "ODE,depvar" "" }{TEXT -1 2 " " }}}} X0u|i@m5!TH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-%\"FG6%%\"tG %\"yG%\"zGF/F0F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diffv ars(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'-%\"FG6%%\"tG%\"yG%\"zGF )F(F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff( op(\") ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-F$6$-F$6$-F$6$-%\"FG6%% \"tG%\"yG%\"zGF/F0F0F1" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Conta ct the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nA stronomical Applications Dept.\nU.S. Naval Observatory\n3450 Massachus etts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 " murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http:/ /riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPET=oderoutindiffvarsyntaxexprdiffpurposreturnvariablderivatquantitlistformvarvarndependindependorderhencexamplproducoprecovoriginalargumentcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitdepvar[utilitroutinpulloutsyntaxexprsubexprnamefunctlistpurposselectivefactoroutsubexpressexpressexamplbelowargumentfoonotealwaycheckanswwecanstackexecutpullnoticordersignificantrepeatbutcasetryinghadnoeffectanothsequenccourshavedonestepslightmorecomplicatformresultdifferenctoopsstillcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsotermfuncfuncopntactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoutilitdepvarTdivisorTdn'done[drGPQ''K4M4N4O4TTTTT[beffect[elapsPenclos TT encounter'enterK4entirequatQ''O4TTT?p  T@argu differenct[ differential O477diffexpr M487difford M487diffvar87TTdivisorYdn'doesYdone77Y[drWPQ'K4M4N4O47787TTTTYY[byyeffect[elapsPenclos 87Y encounter77enterK4entirepsilonYequatQ'O47787TYket 87Ybranch77bugYbut[callP   orderM4TTYY[yoriginal'77T(;still[storPstring PYstructur'studQsubexpr Y[ subexpress[subsO4subslistO4suchYsuitablT\")*&,*F&#F*FL*(,(F5FOF7#F1\"#;F;#F*\"$#> F*F)F*F?F*F*F@FOFC#F1\"#7F*FGF,F***F'F(F)F.F+FJFGF.#F,FL*&,*F&#FDF \+ " 0 "" {MPLTEXT 4 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 128 1 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "D ate" 256 257 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" 256 260 1 {CSTYLE "" -1 -1 "Verdana" 1 11 0 0 128 1 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 262 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 262 "" 0 "" {HYPERLNK 17 "Utility Routines" 2 "utilities " "" }{TEXT -1 11 " \226 sinfix()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 263 "" 0 "" {TEXT 259 8 "sinfix( " }{TEXT 261 53 "e xpr::\{algebraic,array(1),`=`\}, var::\{algebraic,list\}" }{TEXT 260 2 " )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 6 "Force " }{XPPEDIT 18 0 "1-sin^2" ",&\"\"\"\"\"\"*$%$sin G\"\"#!\"\"" }{TEXT -1 5 " ==> " }{XPPEDIT 18 0 "cos^2" "*$%$cosG\"\"# " }{TEXT -1 2 ". " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" } }{PARA 15 "" 0 "" {TEXT 257 4 "expr" }{TEXT -1 32 " Expression to b e simplified." }}{PARA 15 "" 0 "" {TEXT 258 9 "var " }{TEXT -1 49 "The argument of the cosine functions to simplify." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "-rho(tau)*diff(theta(tau),tau)^2-rho(tau)*diff(phi(t au),tau)^2-2*rho(tau)*diff(phi(tau),tau)+rho(tau)*sin(theta(tau))^2+di ff(diff(rho(tau),tau),tau)+rho(tau)*diff(phi(tau),tau)^2*sin(theta(tau ))^2+2*rho(tau)*diff(phi(tau),tau)*sin(theta(tau))^2 = (rho(tau)^3-1)/ rho(tau)^2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&-%$rhoG6#%$tauG\" \"\"-%%diffG6$-%&thetaGF(F)\"\"#!\"\"*&F&F*-F,6$-%$phiGF(F)F0F1*&F&F*F 3F*!\"#*&F&F*-%$sinG6#F.F0F*-F,6$-F,6$F&F)F)F**(F&F*F3F0F:F0F**(F&F*F3 F*F:F0F0*&,&*$F&\"\"$F*F1F*F*F&F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sinfix( \", theta(tau), factor );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,**&-%$rhoG6#%$tauG\"\"\"-%%diffG6$-%&thetaGF(F)\"\" #!\"\"-F,6$-F,6$F&F)F)F*F&F**&,(F&F1*&F&F*-F,6$-%$phiGF(F)F0F1*&F&F*F9 F*!\"#F*-%$cosG6#F.F0F**&,&*$F&\"\"$F*F1F*F*F&F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "termfunc( lhs(\"), factor ) = rhs(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&-%$rhoG6#%$tauG\"\"\"-%%diffG6$-% &thetaGF(F)\"\"#!\"\"-F,6$-F,6$F&F)F)F*F&F**(F&F*,&-F,6$-%$phiGF(F)F*F *F*F0-%$cosG6#F.F0F1*&,&*$F&\"\"$F*F1F*F*F&!\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "collect(\",rho(tau));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,(*$-%%diffG6$-%&thetaG6#%$tauGF.\"\"#!\"\"\"\"\"F 1*&,&-F)6$-%$phiGF-F.F1F1F1F/-%$cosG6#F+F/F0F1-%$rhoGF-F1F1-F)6$-F)6$F ;F.F.F1,&F;F1*$F;!\"#F0" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Cont act the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\n Astronomical Applications Dept.\nU.S. Naval Observatory\n3450 Massachu setts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http: //riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "utilities" 2 "utilities" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "cosfix" 2 "utilities,cosfix" "" } {TEXT -1 1 " " }}}} icolon(sensTsequenc[setdiffG6$-% &thetaGF(F)\"\"#!\"\"-F,6$-F,6$F&F)F)F*F&F**(F&F*,&-F,6$-%$phiGF(F)F*F *F*F0-%$cosG6#F.F0F1*&,&*$F&\"\"$F*F1F*F*F&!\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT ySutilitroutinsinfixsyntaxexpralgebraicarravarlistpurposforcargumentexpresssimplifiargumcosinfunctionsimplifexamplrhotaudiffthetaphisinfactortermfunclhsrhscollectcontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsocosfixԁ ScatJ-=F ScoeffJ y4 ScoeffsJKd Sconvert SdebugoptsJʞ.h3覃ċ SdegreeJ-R<܂ Sdenom 4Jb  SdiffJ&`gBT SdisassembleJKh5it SdivideJ> S$J'胴6[L SentriesJ SevalJȃD Sevalb 1 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" 256 258 1 {CSTYLE "" -1 -1 "Verdana" 1 11 0 0 128 1 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Address" 0 259 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date " 259 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 261 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 263 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 264 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 265 1 {CSTYJy8{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 128 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 128 0 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 177;{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 0 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 0 255 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Non-prop ortional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 0 128 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 128 1 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 256 257 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 4 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Address - email" 256 258 1 {CSTYLE "" -1 -1 "Verdana" 1 11 0 0 128 1 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "A ddress" 0 259 1 {CSTYLE "" -1 -1 "" 1 11 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 1 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Date" 259 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 261 1 {CSTYLE "" -1 -1 "Courier" 0 10 128 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 263 "" 0 "" {HYPERLNK 17 "ODE Routines" 2 "ODE" "" } {TEXT -1 17 " \226 indvarchange()" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 258 72 "indvarchange( deqn, Teqn::e quation, t::name, T::name <, unknowns::set> )" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 15 "" 0 "" {TEXT -1 35 "Transform ol d independent variable " }{TEXT 259 1 "t" }{TEXT -1 29 " to new indepe ndent variable " }{TEXT 260 1 "T" }{TEXT -1 30 " in the differential e quation " }{TEXT 261 4 "deqn" }{TEXT -1 3 ". " }{TEXT 262 4 "Teqn" } {TEXT -1 40 " must be an equation of the form \n " }{XPPEDIT 18 0 "g(T) = f(t)" "/-%\"gG6#%\"TG-%\"fG6#%\"tG" }{TEXT -1 7 "\nwhere " } {TEXT 264 1 "T" }{TEXT -1 34 " is the new independent variable, " } {TEXT 263 1 "f" }{TEXT -1 21 " is some function of " }{TEXT 265 1 "t" }{TEXT -1 6 ", and " }{TEXT 284 1 "g" }{TEXT -1 21 " is some function \+ of " }{TEXT 283 1 "T" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT -1 29 "An allowable special case is " }{XPPEDIT 18 0 "Teqn := T=t" ">%%TeqnG /%\"TG%\"tG" }{TEXT -1 33 ". See the second Example below. " }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Arguments" }}{PARA 15 "" 0 "" {TEXT 257 4 "deqn" }{TEXT -1 53 " An expression or equation contain ing derivatives." }}{PARA 15 "" 0 "" {TEXT 266 1 "t" }{TEXT -1 43 " \+ The old independent variable in " }{TEXT 268 4 "deqn" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 287 1 "T" }{TEXT -1 39 " Th e new independent variable." }}{PARA 15 "" 0 "" {TEXT 267 4 "Teqn" } {TEXT -1 51 " An equation specifying the transformation from " } {TEXT 269 1 "t" }{TEXT -1 33 " to the new independent variable " } {TEXT 289 1 "T" }{TEXT -1 40 ". \n It must be of the form " }{XPPEDIT 18 0 "g(T) = f(t)" "/-%\"gG6#%\"TG-%\"fG6#%\"tG" }{TEXT -1 8 ", where " }{TEXT 270 1 "T" }{TEXT -1 33 " is the new independent variable." }}{PARA 15 "" 0 "" {TEXT 271 8 "unknowns" }{TEXT -1 227 " \+ A set of functions which are to be treated as \"unknown\" functions \+ so that the \n old independent variable is simply r eplaced by the new independent variable \n rather t han the solution of " }{TEXT 272 4 "Teqn" }{TEXT -1 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 15 "" 0 "" {TEXT 273 17 "A simple example." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Tran sform " }{TEXT 277 1 "t" }{TEXT -1 34 " according to the simple functi on " }{XPPEDIT 18 0 "theta = omega*t" "/%&thetaG*&%&omegaG\"\"\"%\"tGF &" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode : = A*diff(x(t),t,t) + B*diff(x(t),t) - C = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&%\"AG\"\"\"-%%diffG6$-F+6$-%\"xG6#%\"tGF2F2 F)F)*&%\"BGF)F-F)F)%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "teqn := theta=omega*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%teqnG/%&thetaG*&%&omegaG\"\"\"%\"tGF)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "indvarchange(ode,teqn,t,theta);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,(*(%\"AG\"\"\"%&omegaG\"\"#-%%diffG6$-F+6$-%\" xG6#%&thetaGF2F2F'F'*(%\"BGF'F(F'F-F'F'%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 15 "" 0 "" {TEXT 290 24 "A second simple example." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Suppose that instead we wish to let " } {TEXT 278 1 "t" }{TEXT -1 47 " be replaced by some unspecified functio n, say " }{XPPEDIT 18 0 "phi(t)" "-%$phiG6#%\"tG" }{TEXT -1 27 ". Thi s is done by setting " }{XPPEDIT 18 0 "Teqn := phi = t" ">%%TeqnG/%$ph iG%\"tG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "indvarchange( ode, phi=t, t, phi );" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,(*&%\"AG\"\"\",&*&-%%diffG6$-F+6$-%$phiG6#%\"tGF2F2F'-F+6$-%\"xG6# F0F0F'F'*&F-\"\"#-F+6$F3F0F'F'F'F'*(%\"BGF'F-F'F3F'F'%\"CG!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 274 27 "A more complicated example." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "This artificially gnarly equation is chosen specifically to fully \+ exercise indvarchange." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 " ode := A(t)*cos(t)*diff(x(t),t,t) + \n B(t)*x(t)*(2*t/(x(t)+t)-1 /(x(t)-t))*sin(t)*cos(t)*diff(x(t),t)^2 \n - C(t) - t*(A(t)-B(t) )*tan(t)*diff(F(t),t)^2*diff(x(t),t) \n = alpha*t*sin(t);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$odeG/,**(-%\"AG6#%\"tG\"\"\"-%$cosG F*F,-%%diffG6$-F06$-%\"xGF*F+F+F,F,*.-%\"BGF*F,F4F,,&*&F+F,,&F4F,F+F,! \"\"\"\"#*$,&F4F,F+F " 0 "" {MPLTEXT 1 0 59 "indvarchange( ode, teqn, t, \+ theta, \{A(t),B(t),C(t),x(t)\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ ,***-%\"AG6#%&thetaG\"\"\"-%$cosG6#*&F)F*%&omegaG!\"\"F*F/\"\"#-%%diff G6$-F36$-%\"xGF(F)F)F*F**0-%\"BGF(F*F7F*,&*(F)F*F/F0,&F7F*F.F*F0F1*$,& F7F*F.F0F0F0F*-%$sinGF-F*F+F*F/F1F5F1F*-%\"CGF(F0*.F)F*,&F&F*F:F0F*-%$ tanGF-F*F/F1-F36$-%\"FGF(F)F1F5F*F0**%&alphaGF*F)F*F/F0FAF*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Check the transformation by transf orming back to the original ODE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "indvarchange( \", teqn, theta, t, \n \{A (theta),B(theta),C(theta),x(theta)\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**(-%\"AG6#%\"tG\"\"\"-%$cosGF(F*-%%diffG6$-F.6$-%\"xGF(F)F)F* F**.-%\"BGF(F*F2F*,&*&F)F*,&F2F*F)F*!\"\"\"\"#*$,&F2F*F)F:F:F:F*-%$sin GF(F*F+F*F0F;F*-%\"CGF(F:*,F)F*,&F&F*F5F:F*-%$tanGF(F*-F.6$-%\"FGF(F)F ;F0F*F:*(%&alphaGF*F)F*F>F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(\"-ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 275 34 " A more complicated transformation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "teqn := theta = omega*sin(t+alpha)+beta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%teqnG/%&thetaG,&*&%&omegaG\"\"\"-%$sinG6#,&% \"tGF*%&alphaGF*F*F*%%betaGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "indvarchange( ode, teqn, t, theta, \{A(t),B(t),C(t),x(t)\} );" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**(-%\"AG6#%&thetaG\"\"\"-%$cosG6#, &%&alphaG!\"\"-%'arcsinG6#*&,&F)F0%%betaGF*F*%&omegaGF0F0F*,&*(F7F*-%$ sinG6#,$F1F0F*-%%diffG6$-%\"xGF(F)F*F0*(F7\"\"#-F,FF)F*F*F*F **2-%\"BGF(F*FAF*,&*&F.F*,(FAF*F/F0F1F0F0FD*$,(FAF*F/F*F1F*F0F0F*-F;F- F*F+F*F7FDFEFDF>FDF*-%\"CGF(F0*0F.F*,&F&F*FIF0F*-%$tanGF-F*F7\"\"$FEFW -F?6$-%\"FGF(F)FDF>F*F0*(F/F*F.F*FPF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "collect( normal(\"),\n [diff(x(theta),theta$2 ),diff(x(theta),theta)], \n factor );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**,-%$cosG6#,&%&alphaG\"\"\"-%'arcsinG6#*&,&%&thetaG! \"\"%%betaGF+F+%&omegaGF2F+F+-%\"AG6#F1F+,(F3F+F1F2F4F+F+,(F3F2F1F+F4F +F+-%%diffG6$-F;6$-%\"xGF7F1F1F+F+*6-%\"BGF7F+F?F+-%$sinGF(F+F&F+F8F+F 9F+,2*&F*F+F?F+\"\"#*$F*FHFH*&F*F+F,F+\"\"%*&F,F+F?F+FH*$F,FHFHF?F+F*F 2F,F2F+,(F?F2F*F+F,F+F2,(F?F+F*F+F,F+F2F=FHF2*&,.*.F,F+-%$tanGF(F+F4\" \"$*(F8F+F9F+F4!\"##FUFH-F;6$-%\"FGF7F1FHF5F+F2*.F,F+FSF+F4FUFVFXFYFHF BF+F+*.FSF+F4FUFVFXFYFHF*F+FBF+F+*(F5F+F&F+F3F+F+*(F5F+F&F+F1F+F2*.FSF +F4FUFVFXFYFHF*F+F5F+F2F+F=F+F+-%\"CGF7F2*(F*F+F)F+FDF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Check the transformation by transforming \+ back to the original ODE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "indvarchange( \", teqn, theta, t, \n \{A(theta),B(the ta),C(theta),x(theta)\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**,-%$ cosG6#,&%&alphaG\"\"\"-%'arcsinG6#,$-%$sinG6#,&%\"tGF+F*F+!\"\"F+F+-% \"AG6#F4F+,&*&%&omegaGF+F0F+F5F;F+F+,&F:F+F;F+F+,&**F;!\"#,&F+F+*$F0\" \"#F5#!\"$FB-%%diffG6$-%\"xGF8F4F+F0F+F+*(F;F?F@F5-FF6$FEF4F+F+F+F+*:- %\"BGF8F+FHF+-F1F(F+F&F+F9F+F " 0 "" {MPLTEXT 1 0 25 "simplify(\"-ode,symb olic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 276 92 "If a nonunique \+ solution is encountered for t, the routine will prompt for a solution \+ branch." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "teqn := theta = \+ omega*t^2+alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%teqnG/%&thetaG, &*&%&omegaG\"\"\"%\"tG\"\"#F*%&alphaGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "indvarchange( ode, teqn, t, theta, \{A(t),B(t),C(t),x (t)\} );" }}{PARA 6 "" 1 "" {TEXT -1 50 "indvarchange/solve[132]: nonu nique solution for t:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&%&omegaG!\" \"*&F$\"\"\",&%&thetaGF'%&alphaGF%F'#F'\"\"#,$F#F%" }}{PARA 6 "" 1 "" {TEXT -1 80 "indvarchange/solve[132]: choose a branch, followed by a s emicolon (e.g., \"2;\"): " }}}{EXCHG {PARA 0 "==> " 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&omegaG!\"\"*&F$\"\"\",&%& thetaGF'%&alphaGF%F'#F'\"\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**(- %\"AG6#%&thetaG\"\"\"-%$cosG6#*&%&omegaG!\"\"*&F/F*,&F)F*%&alphaGF0F*# F*\"\"#F*,&*&F/F*-%%diffG6$-%\"xGF(F)F*F5*(F/F*F2F*-F96$F8F)F*\"\"%F*F **2-%\"BGF(F*F;F*,&*(F/F0F1F4,&F;F*F.F*F0F5*$,&F;F*F.F0F0F0F*-%$sinGF- F*F+F*F/F*F2F*F8F5F@-%\"CGF(F0*.F2F5,&F&F*FBF0F*-%$tanGF-F*F/F*-F96$-% \"FGF(F)F5F8F*!\")**F3F*F/F0F1F4FIF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Check the transformation by transforming back to the original O DE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "indvarchange( \", te qn, theta, t, \n \{A(theta),B(theta),C(theta),x(theta)\} \+ );" }}{PARA 6 "" 1 "" {TEXT -1 50 "indvarchange/solve[132]: nonunique \+ solution for t:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&%&omegaG!\"\"*&F$ \"\"\",&%&thetaGF'%&alphaGF%F'#F'\"\"#,$F#F%" }}{PARA 6 "" 1 "" {TEXT -1 80 "indvarchange/solve[132]: choose a branch, followed by a semicol on (e.g., \"2;\"): " }}}{EXCHG {PARA 0 "==> " 0 "" {MPLTEXT 1 0 2 "1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&omegaG!\"\"*&F$\"\"\",&%&theta GF'%&alphaGF%F'#F'\"\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**(-%\"AG 6#%\"tG\"\"\"-%$cosG6#*&%&omegaG!\"\"*&F/\"\"#F)F2#F*F2F*,&*(F/F*F1#F0 F2-%%diffG6$-%\"xGF(F)F*F**(F/F2F)F2,&*(F1#!\"$F2F7F*F/F*#F0\"\"%*(F/! \"#F)FD-F86$F7F)F*#F*FBF*FBF*F**.-%\"BGF(F*F:F*,&*(F/F0F1F3,&F:F*F.F*F 0F2*$,&F:F*F.F0F0F0F*-%$sinGF-F*F+F*F7F2F*-%\"CGF(F0*0F/F*F)F2,&F&F*FI F0F*-%$tanGF-F*-F86$-%\"FGF(F)F2F1F6F7F*F0**%&alphaGF*F/F0F1F3FPF*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(\"-ode,symbolic); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.S. Naval Obse rvatory\n3450 Massachusetts Ave., NW\nWashignton, DC 20392" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}{PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 " ODE" 2 "ODE" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "utilities" 2 "utilitie s" "" }}}} ǃgcJ ǃgenpolyJ{ ǃgetuserinterfaceJ   ǃgotoJlyH ǃhas 0F1F3,&F:F*F.F*F 0F2*$,&F:F*0 0 0 -1 0 }{PSTYLE "" 15 270 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 271 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 272 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 273 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 274 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 275 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 276 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 277 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 278 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 279 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 280 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 281 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 282 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 284 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 285 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 286 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 287 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 288 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 289 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 290 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 291 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 257 16 "Utility Routines" }}{PARA 0 "" 0 "" {TEXT -1 131 "These utility routines are located in the file util s.p. This file must be read in before using any of these routines. Fo r example," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "read`c:/maple/ utils.p`;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "Syntax" }}{PARA 0 " " 0 "" {TEXT 20 60 "NOTE: Arguments enclosed in angle brackets <> are optional." }}{PARA 276 "" 0 "" {TEXT -1 48 "annular_average( expr, co ords, a, b, paramlist )" }}{PARA 273 "" 0 "" {HYPERLNK 17 "applyfunc" 2 "utilities,applyfunc" "" }{TEXT 266 50 "( expr, func::\{procedure,fu nction\}, n::nonnegint )" }}{PARA 273 "" 0 "" {TEXT -1 70 "CombineTrig ( expr::\{algebraic,`=`\}, var1::algebraic, var2::algebraic )" }} {PARA 275 "" 0 "" {TEXT -1 30 "`CombineTrig/optimize`( expr )" }} {PARA 286 "" 0 "" {TEXT -1 24 "CombineTrigOpt( expr ) " }{TEXT 262 22 "(this is an alias for " }{TEXT -1 22 "`CombineTrig/optimize`" } {TEXT 263 1 ")" }}{PARA 281 "" 0 "" {TEXT -1 56 "common_factor( expr:: algebraic, varlist::\{list,string\} )" }}{PARA 290 "" 0 "" {HYPERLNK 17 "cosfix" 2 "utilities,cosfix" "" }{TEXT -1 57 "( expr::\{algebraic, array(1),`=`\}, var::\{algebraic,list\} )" }}{PARA 268 "" 0 "" {HYPERLNK 17 "csquare" 2 "utilities,csquare" "" }{TEXT -1 8 "( expr:: " }{TEXT 259 42 "\{algebraic,array,indexed,equation,numeric\}" }{TEXT 261 2 " \n" }{TEXT -1 17 " <, var::" }{TEXT 260 36 "\{name,set ,list,`+`,`*`,`^`,function\}" }{TEXT -1 3 "> )" }}{PARA 269 "" 0 "" {TEXT 258 51 "debug_print( pname, msg, verbosity_level <, expr> )" }} {PARA 282 "" 0 "" {HYPERLNK 17 "expansion" 2 "utilities,expansion" "" }{TEXT -1 140 "( expression::\{list,set,'array'(1),'array'(2),matrix,` =`,algebraic\},\n params::\{set,name,list,function\}, expans ion_order::integer )" }}{PARA 279 "" 0 "" {HYPERLNK 17 "funcops" 2 "ut ilities,funcops" "" }{TEXT -1 54 "( expr::`+`, func::procedure, oplist ::\{integer,list\} )" }}{PARA 274 "" 0 "" {TEXT -1 21 "FunctionCalls( \+ expr )" }}{PARA 265 "" 0 "" {TEXT -1 38 "getsqrts( expr::\{algebraic,e quation\} )" }}{PARA 277 "" 0 "" {TEXT -1 21 "is_real_parts( expr )" } }{PARA 289 "" 0 "" {HYPERLNK 17 "location" 2 "utilities,location" "" } {TEXT -1 34 "( expr::anything, term::anything )" }}{PARA 280 "" 0 "" {TEXT -1 51 "polytest( expr::algebraic, varlist::\{list,string\} )" }} {PARA 288 "" 0 "" {HYPERLNK 17 "pullout" 2 "utilities,pullout" "" } {TEXT -1 66 "( expr::\{`+`,`^`,`*`\}, subexpr::\{name,function,`+`,`*` ,`^`,list\} )" }}{PARA 272 "" 0 "" {TEXT -1 98 "rootfunc( expr::\{alge braic,equation,list,set,indexed,array\}, \n func::procedure < , args> )" }}{PARA 278 "" 0 "" {TEXT -1 41 "RotAxis( theta::algebraic, axis::string )" }}{PARA 266 "" 0 "" {TEXT -1 27 "signsqrt( expr::\{`* `,`^`\} )" }}{PARA 291 "" 0 "" {HYPERLNK 17 "sinfix" 2 "utilities,sinf ix" "" }{TEXT -1 57 "( expr::\{algebraic,array(1),`=`\}, var::\{algebr aic,list\} )" }}{PARA 267 "" 0 "" {TEXT -1 100 "small_divisors( expr:: \{algebraic,list(algebraic),array(1)\},\n varlist::\{li st,string\} )" }}{PARA 287 "" 0 "" {HYPERLNK 17 "termfunc" 2 "utilitie s,termfunc" "" }{TEXT 265 19 "( expr::\{algebraic," }{TEXT -1 8 "array (1)" }{TEXT 264 20 "\}, func::procedure )" }}{PARA 270 "" 0 "" {HYPERLNK 17 "tmsg" 2 "utilities,tmsg" "" }{TEXT -1 25 "( proc_name, m sg <, t0> )" }}{PARA 285 "" 0 "" {TEXT -1 38 "topsqrts( expr::\{algebr aic,equation\} )" }}{PARA 284 "" 0 "" {TEXT -1 30 "topsqrt( expr::\{`+ `,`*`,`^`\} )" }}{PARA 271 "" 0 "" {TEXT -1 76 "`type/fractional_power `( expr::\{algebraic,equation,list,set,indexed,array\} )" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" {TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU. S. Naval Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 2039 2" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }} {PARA 15 "" 0 "" {TEXT 256 37 "http://riemann.usno.navy.mil/murison/" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {HYPERLNK 17 "AESOP" 2 "AESOP" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ODE " 2 "ODE" "" }{TEXT -1 1 " " }}}} " {HYPERLNK 17 "tmsg" 2 "utilities,tmsg" "" }{TEXT -1 25 "( proc_name, m sg <, t0> )" }}{PARA 285 "" 0 "" {TEXT -1 38 "topsqrts( expr::\{algebr aic,equation\} )" }}{PARA 284 "" 0 "" {TEXT -1 30 "topsqrt( expr::\{`+ `,`*`,`^`\} )" }}{PARA 271 "" 0 "" {TEXT -1 76 "`type/fractional_power `( expr::\{algebraic,equation,list,set,indexed,array\} )" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 ""YutilitroutintheslocatfileutilmustreadbeforusinganyexamplmaplsyntaxnoteargumentenclosanglbracketoptionalannularaveragexprcoordparamlistapplyfuncfuncprocedurfunctnonnegintcombinetrigalgebraicvaroptimizcombinetrigoptaliacommonfactorvarlistliststringcosfixarracsquarindexequatnumericnamesetdebugprintpnammsgverbositlevelexpansexpressmatrixparamorderintegfuncopoplistfunctioncallgetsqrtrealpartanythtermpolytestpulloutsubexprrootfuncargsrotaxithetaaxissignsqrtsinfixsmalldivisortermfunctmsgproctopsqrttypefractionalpowercontactauthordrmarcmurisonastronomicalapplicatdeptnavalobservatormassachusettavenwwashigntondcriemanusnonavymilhttpalsoaesopodeyyynavyW*PQ'K4M4N4O47787TTTTY[byyyndQnew77no Y[nonintegY nonnegintYbynonuniqu77nops