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{SECT 0 {PARA 258 "" 0 "" {TEXT 256 10 "Chapter 16" }}{PARA 259 "" 0 "
" {TEXT 257 17 "Solving Equations" }}{SECT 1 {PARA 3 "" 0 "Basics" 
{TEXT -1 29 "16.1 Equations in One Unknown" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "eqn := (x-1)*(x^2+x+1);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq
nG*&,&%\"xG\"\"\"!\"\"F(F(,(*$)F'\"\"#\"\"\"F(F'F(F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sol := solve( eqn, x );  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$solG6%\"\"\",&#!\"\"\"\"#F&*&%\"IGF&-%%sq
rtG6#\"\"$\"\"\"#F&F*,&F(F&F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "subs( x=sol[2], eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&
#!\"$\"\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"#F(F'F(,(*$),&#!\"\"F'F(
F)F0F'F/F(F0F(F)F0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "exp
and(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "eval( eqn, x=sol[3] );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&#!\"$\"\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"#!\"
\"F'F(,(*$),&F0F(F)F0F'F/F(#F(F'F(F)F0F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn := x^3 + 2*a*x^2 + a*
x = 1;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*$)%\"xG\"\"$\"
\"\"\"\"\"*&%\"aGF,)F)\"\"#F+F0*&F.F+F)F,F,F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "solve( eqn, x );  " }}{PARA 12 "" 1 "" {XPPMATH 
20 "6%,(*$),**$)%\"aG\"\"#\"\"\"\"#s\"$3\"\"\"\"*$)F)\"\"$F+!#k*$-%%sq
rtG6#,*F/!#%)*$)F)\"\"%F+!#7F'F-\"#\")F.F+\"#7#F.F1F+#F.\"\"'*&,&F)F?F
'#!\"%\"\"*F+*$)F&#\"\"\"F1F+!\"\"!\"'F)#!\"#F1,*F$#!\"\"F>FBF1F)FM*(%
\"IGF.-F56#F1F+,&F$F@FBFAF.#F.F*,*F$FPFBF1F)FMFR#FQF*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rationalize( %2^3 );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,**$)%\"aG\"\"#\"\"\"#!\"\"\"$['#F*\"$K%\"\"\"*$
)F&\"\"$F(#F.\"$H(*$-%%sqrtG6#,*F/!#%)*$)F&\"\"%F(!#7F$\"$3\"\"#\")F.F
(#F.\"%))Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "Abbreviations" {TEXT -1 27 "16.2 Abbreviations in Solve
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve( a + ln(x-3) - ln(x), x );  \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#%\"aG\"\"\",&!\"\"\"
\"\"F%F,!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve
( a + ln(x-3) - ln(x) );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG
,&-%#lnG6#,&%\"xG\"\"\"!\"$F,!\"\"-F(6#F+F,/F+F+" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 44 "x^7 - 2*x^6 - 4*x^5 - x^3 + x^2 + 6*x + 4;  \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG\"\"(\"\"\"\"\"\"*$)F&\"
\"'F(!\"#*$)F&\"\"&F(!\"%*$)F&\"\"$F(!\"\"*$)F&\"\"#F(F)F&F,\"\"%F)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "solve(%); " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%,&\"\"\"F$*$-%%sqrtG6#\"\"&\"\"\"F$,&F$F$F%!\"\"-%
'RootOfG6#,(*$)%#_ZGF)F*F$F3F,F,F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "Difficulties" {TEXT -1 22 
"16.3 Some Difficulties" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve( cos(x) = x
, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,&%#_ZG\"\"\"-%$
cosG6#F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "solve( x = x + 1, x );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "
solve( cos(x) = x^2, x ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "_SolutionsMayBeLost;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "readlib(forget):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "forget( solve ):  # forget previous results of solve
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "infolevel[solve] := 2: \+
# make Maple more communicative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "solve( cos(x) = x^2, x );  " }}{PARA 6 "" 1 "" {TEXT -1 36 "so
lve:   Warning: no solutions found" }}{PARA 6 "" 1 "" {TEXT -1 46 "sol
ve:   Warning: solutions may have been lost" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 40 "infolevel[solve] := 1:  # reset userinfo" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := x + x^(1/3) = -2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&%\"xG\"\"\"*$)F'#F(\"\"$\"\"
\"F(!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve( eqn, x )
;" }}{PARA 6 "" 1 "" {TEXT -1 36 "solve:   Warning: no solutions found
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert( eqn, RootOf );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"-%'RootOfG6#,&*$)%#_
ZG\"\"$\"\"\"F&F%!\"\"F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "solve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\",&#!\"&\"\"#
\"\"\"*&%\"IGF(-%%sqrtG6#\"\"(\"\"\"#F(F',&F%F(F)#F#F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "solve( sin(x) = 3*x/Pi, x );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RootOfG6#,&%#_ZG\"\"$*&-%$sinG6#F'\"\"\"%#PiGF-!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%);  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "(x^6-1)^x = 0; solve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/),&*$)%\"xG\"\"'\"\"\"\"\"\"!\"\"F+F(\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"\",&#F#\"\"#F#*&%\"IGF#-%%sqrtG6#\"\"$\"\"\"#!\"\"F
&,&F%F#F'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x + x^(1/2)+
x^(1/3)+x^(1/4) = 4;  solve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
,*%\"xG\"\"\"*$-%%sqrtG6#F%\"\"\"F&*$)F%#F&\"\"$F+F&*$)F%#F&\"\"%F+F&F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "x^3*(ln(5)-ln(x))=3;  solve(%,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&)%\"xG\"\"$\"\"\",&-%#lnG6#\"\"&\"\"\"-F+6#F&!
\"\"F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$-%$expG6#,$-%)LambertWG6
##!\"*\"$D\"#\"\"\"\"\"$\"\"&,$-F%6#,$-F)6$!\"\"F+F.F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "arccos(3*x) = 2*arcsin(x);  solve(%
,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arccosG6#,$%\"xG\"\"$,$-%'
arcsinG6#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#!\"$\"\"%\"\"\"
*$-%%sqrtG6#\"#<\"\"\"#F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "arccos(2*x) = arctan(3*x); solve(%,x );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'arccosG6#,$%\"xG\"\"#-%'arctanG6#,$F(\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#,&!\"#\"\"\"*$-F&6#\"#5
\"\"\"\"\"#F/#F*\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ma
x( x, 2*x-2 ) = min( x^2-1, 5-x ); solve(%,x );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$maxG6$%\"xG,&F'\"\"#!\"#\"\"\"-%$minG6$,&*$)F'F)\"
\"\"F+!\"\"F+,&\"\"&F+F'F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&#\"\"
\"\"\"#F%*$-%%sqrtG6#\"\"&\"\"\"#!\"\"F&,&F$F%F'F$#\"\"(\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf([%]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%$!+!*)R.='!#5$\"+*)R.=;!\"*$\"+LLLLBF)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot( \{ max( x, 2*x-2 ), mi
n( x^2-1, 5-x ) \}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "  x = -4..4 \+
); " }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7U7$$!\"%\"\"!F(7$$!1n
mmmFiDQ!#:F,7$$!1LLLo!)*Qn$F.F07$$!1nmmwxE.NF.F37$$!1mmmOk]JLF.F67$$!1
MLL[9cgJF.F97$$!1nmmhN2-IF.F<7$$!1+++N&oz$GF.F?7$$!1nmm\")3DoEF.FB7$$!
1+++:v2*\\#F.FE7$$!1LLL8>1DBF.FH7$$!1nmmw))yr@F.FK7$$!1+++S(R#**>F.FN7
$$!1++++@)f#=F.FQ7$$!1+++gi,f;F.FT7$$!1nmm\"G&R2:F.FW7$$!1LLLtK5F8F.FZ
7$$!1MLL$HsV<\"F.Fgn7$$!1-++]&)4n**!#;Fjn7$$!1PLLL\\[%R)F\\oF^o7$$!1)*
****\\&y!pmF\\oFao7$$!1******\\O3E]F\\oFdo7$$!1KLLL3z6LF\\oFgo7$$!1MLL
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pfKF\\oFdp7$$\"1HLLLm&z\"\\F\\oFgp7$$\"1(******z-6j'F\\oFjp7$$\"1#****
**4#32$)F\\oF]q7$$\"1%*****\\#y'G**F\\oF`q7$$\"1******H%=H<\"F.Fcq7$$
\"1mmm1>qM8F.Ffq7$$\"1++++.W2:F.Fiq7$$\"1LLLep'Rm\"F.F\\r7$$\"1+++S>4N
=F.F_r7$$\"1LL$ed*f:>F.Fbr7$$\"1mmm6s5'*>F.Fer7$$\"1LLL))3E!3#F.$\"1lm
mw<_g@F.7$$\"1+++lXTk@F.$\"1******H\"H)GBF.7$$\"1mmmmd'*GBF.$\"1LLLL:$
zl#F.7$$\"1+++DcB,DF.$\"1******\\7Z-IF.7$$\"1MLLt>:nEF.$\"1nmmYRIMLF.7
$$\"1LLL.a#o$GF.$\"1mmm13ltOF.7$$\"1nmm^Q40IF.$\"1LLL.x=5SF.7$$\"1+++!
3:(fJF.$\"1+++g,V>VF.7$$\"1nmmc%GpL$F.$\"1LLL8p&Qn%F.7$$\"1LLL8-V&\\$F
.$\"1mmmE/'3*\\F.7$$\"1+++XhUkOF.$\"1+++!H_)G`F.7$$\"1+++:o<EQF.$\"1++
+ION_cF.7$$\"\"%F*$\"\"'F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-F$6$7W7$F($
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<$\"1mmmhN2-!)F.7$$!1LLL[5-?HF.$\"1(p4F#H_EvF.7$F?$\"1Y+l0a1aqF.7$FB$
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plF_p7$F^o$!1gRSqAE`HF\\o7$Fao$!1\"*HRH\"RBb&F\\o7$Fdo$!1o-K9$[QZ(F\\o
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1?Kelv%Gg&F\\o7$F]q$!1tfQ)pQ#*4$F\\o7$F`q$!1eq(z?[8U\"!#<7$Fcq$\"1KmLM
kPdPF\\o7$Ffq$\"1Sjf'z\"H9yF\\o7$Fiq$\"1!4k!eiPs7F.7$F\\r$\"1%3D%Qgyo<
F.7$F_r$\"1jHDGCcnBF.7$Fbr$\"1XG$\\t@&pEF.7$Fer$\"1swY+SW%)HF.7$Fhr$\"
1nmm6\"R(>HF.7$F]s$\"1+++NaeNGF.7$Fbs$\"1MLLLU.rEF.7$Fgs$\"1+++vVw)\\#
F.7$F\\t$\"1mmmE![GL#F.7$Fat$\"1nmm'fuJ;#F.7$Fft$\"1LLL[h!\\*>F.7$F[u$
\"1+++?\\GS=F.7$F`u$\"1LLLV:2j;F.7$Feu$\"1nmm'ypX]\"F.7$Fju$\"1+++bQdN
8F.7$F_v$\"1+++&=BQ<\"F.7$Fdv$\"\"\"F*-Fiv6&F[wF*F\\wF*-%+AXESLABELSG6
$Q\"x6\"%!G-%%VIEWG6$;F(Fdv%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 0 4 
2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 
0 0 0 8362 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "
abs( x + abs(x+1) ) = 1;  solve(%,x );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$absG6#,&%\"xG\"\"\"-F%6#,&F(F)F)F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"!-%*RealRangeG6$,$%)infinityG!\"\"F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( \{%%\}, \{x\} ); " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$<#/%\"xG\"\"!<#1F%!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "solve( sin(x) = 1/2, x );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "readlib(forget):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "forget( solve ); # forget previous answer" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_EnvAllSolutions := true:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( sin(x) = 1/2, x );" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%#PiG#\"\"\"\"\"'*&F$F&%%_B1|irGF&#
\"\"#\"\"$*&F$\"\"\"%%_Z1|irGF&F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "map( about, indets(%,name) ):" }}{PARA 6 "" 1 "" 
{TEXT -1 3 "Pi:" }}{PARA 6 "" 1 "" {TEXT -1 22 "  is assumed to be: Pi
" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 29 "Origi
nally _Z1, renamed _Z1~:" }}{PARA 6 "" 1 "" {TEXT -1 27 "  is assumed \+
to be: integer" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 29 "Originally _B1, renamed _B1~:" }}{PARA 6 "" 1 "" {TEXT 
-1 31 "  is assumed to be: OrProp(0,1)" }}{PARA 6 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve( exp(x) = 2, x );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"#\"\"\"*(%\"IGF(%#Pi
GF(%$_Z|irGF(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( e
xp(-x) = x, x ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)LambertWG6$%%_
NN1G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := produc
t( x-k, k=1..101 ) = 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
nops( \{ solve( eqn, x ) \} );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "_MaxSols := 200:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "nops( \{ solve( eqn, x ) \+
\} ); # true number of solutions  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"$,\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_EnvAllSolutions \+
:= true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eqn := sin(x)^3
 -13/2*sin(x)^2 + 11*sin(x) = 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$eqnG/,(*$)-%$sinG6#%\"xG\"\"$\"\"\"\"\"\"*$)F)\"\"#F.#!#8F2F)\"#6\"\"
%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve( eqn, x );" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6%,*-%'arcsinG6#\"\"#\"\"\"*&F$F(%%_B1|i
rGF(!\"#*&%#PiGF(%%_Z1|irGF(F'*&F-\"\"\"F*F0F(,(F-#F(\"\"'*&F-F0F*F0#F
'\"\"$*&F-F0F.F0F',*-F%6#\"\"%F(*&F9F(F*F0F+*&F-F0F.F0F'*&F-F0F*F0F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve( eqn, sin(x) );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"##\"\"\"F#\"\"%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "solve( cos(2*x) = cos(x), x );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$,(%#PiG#\"\"#\"\"$*&F$\"\"\"%%_B1|irGF)#!\"%
F'*&F$\"\"\"%%_Z1|irGF)F&,$F-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "solve( cos(2*x) = cos(x), cos(x) );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$cosG6#,$%\"xG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "solve( 2*cos(x)^2 - 1 = cos(x), cos(x) );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$#!\"\"\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 38 "seq( solve( cos(x) = s, x ),  s = % );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$,(%#PiG#\"\"#\"\"$*&F$\"\"\"%%_B1|irGF)#!\"%
F'*&F$\"\"\"%%_Z1|irGF)F&,$*&F$F.F/F.F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "(x-1)^2 / (x^2-1) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&*$),&%\"xG\"\"\"!\"\"F)\"\"#\"\"\"F,,&*$)F(F+F,F)F*F)!\"\"\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "solve(%); " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "subs( x=1, %% ); " }}{PARA 8 "" 1 "" {TEXT -1 23 "Err
or, division by zero" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "lim
it( lhs(%%), x=1 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqns := \{ w +x +y + z = 1, \+
w + y = 0, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  2*w + z = 2, v + z
 = 0 \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "vars := indets(
 eqns );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG<'%\"xG%\"yG%\"zG%
\"vG%\"wG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve( eqns, v
ars );              " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"zG,&%\"w
G!\"#\"\"#\"\"\"/%\"yG,$F'!\"\"/%\"vG,&F'F)F(F*/%\"xG,&F'F)F.F*/F'F'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify( \{v,w\}, eqns, \+
[v,w,x,y,z] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,&%\"zG#!\"\"\"
\"#\"\"\"F),$F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simpli
fy( \{v,w\}, eqns, [v,w,z,y,x] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#<$,&%\"xG\"\"\"!\"\"F&,&F%#F&\"\"#F)F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "solve( eqns, \{v,w,y,z\} );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<&/%\"zG,&%\"xG!\"\"\"\"\"F)/%\"wG,&F'#F)\"\"#F-F)/%\"v
G,&F'F)F(F)/%\"yG,&#F(F.F)F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "solvefor[v,w,y,z]( eqns );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#<&/%\"zG,&%\"xG!\"\"\"\"\"F)/%\"wG,&F'#F)\"\"#F-F)/%\"vG,&F'F)F(F)/%
\"yG,&#F(F.F)F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "Systems" {TEXT -1 25 "16.4 Systems of Equations
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alias( I=I, J=sqrt(-1) ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqns := \{ R[1]*I[1]+R[3]*I[
3]+R[4]*I[4]-V[1]=0, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " R[2]*I[2]
-V[2]-R[3]*I[3]=0, R[5]*I[5]-R[6]*I[6]-V[2]=0, " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 " R[5]*I[5]+R[7]*I[7]-R[4]*I[4]=0, I[1]-I[2]-I[3]=0, \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " I[2]-I[6]-I[8]=0, I[5]+I[6]-I[
7]=0, I[4]+I[7]-I[1]=0, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " I[3]+I
[8]-I[4]-I[5]=0 \};  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<+/,(
&%\"IG6#\"\"\"F+&F)6#\"\"#!\"\"&F)6#\"\"$F/\"\"!/,(F,F+&F)6#\"\"'F/&F)
6#\"\")F/F3/,(&F)6#\"\"&F+F6F+&F)6#\"\"(F/F3/,(&F)6#\"\"%F+FAF+F(F/F3/
,*F0F+F9F+FFF/F>F/F3/,**&&%\"RGF*F+F(F+F+*&&FOF1F+F0F+F+*&&FOFGF+FFF+F
+&%\"VGF*F/F3/,(*&&FOF-F+F,F+F+&FUF-F/FPF/F3/,(*&&FOF?F+F>F+F+*&&FOF7F
+F6F+F/FZF/F3/,(FgnF+*&&FOFBF+FAF+F+FRF/F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "currents := \{ seq( I[i], i=1..8 ) \}:  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "resistors := \{ seq( R[i], i
=1..7 ) \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "voltages := \+
\{ V[1], V[2] \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sol :=
 solve( eqns, currents ):   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "assign( sol ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "I[5]
;  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,8**&%\"RG6#\"\"(\"\"\"&F'6#F
*F*&F'6#\"\"#F*&%\"VGF.F*F***&F'6#\"\"%F*F+\"\"\"F0F6&F'6#\"\"$F*F***F
3F6F+F6F-F6F0F6F***F&F6F7F6F-F6F0F6F***F&F6F+F6F0F6F7F6F***F3F6&F1F,F*
&F'6#\"\"'F*F7F6F***F&F6F-F6F3F6F0F6F***F?F6F3F6F>F6F-F6F***F3F6F7F6F&
F6F0F6F***F3F6F7F6F?F6F0F6F***F3F6F7F6F-F6F0F6F*F6,L**F?F6F3F6F7F6F-F6
F***&F'6#\"\"&F*F&F6F+F6F-F6F***FJF6F3F6F+F6F-F6F***FJF6F3F6F+F6F7F6F*
**F?F6F3F6F+F6F7F6F***F?F6F&F6F+F6F-F6F***F?F6F&F6F+F6F7F6F***F?F6F&F6
F7F6F-F6F***F?F6F3F6F+F6F-F6F***F?F6F-F6F3F6FJF6F***F?F6F7F6F3F6FJF6F*
**F?F6F+F6F7F6FJF6F***F?F6F7F6F-F6FJF6F***F?F6F+F6F-F6FJF6F***FJF6F&F6
F-F6F3F6F***FJF6F3F6F7F6F-F6F***FJF6F&F6F7F6F3F6F***FJF6F&F6F+F6F7F6F*
**FJF6F&F6F7F6F-F6F***F?F6F&F6F-F6F3F6F***F?F6F&F6F7F6F3F6F*!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "relations := \{ " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 55 "  A = R[1]*R[2]*R[4] + R[1]*R[3]*R[4] + R[2
]*R[3]*R[4]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "  B = R[5]*R[6] + R
[5]*R[7] + R[6]*R[7], C = R[1]*R[2] " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "      + R[1]*R[3] + R[2]*R[3] + R[2]*R[4] + R[3]*R[4]," }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "  D = R[4]*R[6] \}:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "I[5] := map( simplify, I[5], relations, " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "             [op(resistors),A,B,C,D
] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"IG6#\"\"&*&,**(&%\"VG6#
\"\"\"F.&%\"RG6#\"\"#F.%\"DGF.F.*&&F,F1F.%\"AGF.F.*(F5\"\"\"&F06#\"\"(
F.%\"CGF.F.*(,&F+F.F5F.F.&F06#\"\"$F.F3F8F.F8,(*&F<F8%\"BGF.F.*&&F0F&F
.F6F8F.*&&F06#\"\"'F.F6F8F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "I[5] := map( collect, I[5], [V1,V2,D] );  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"IG6#\"\"&*&,(*&,&*&&%\"VG6#\"\"\"F0&%\"
RG6#\"\"#F0F0*&,&F-F0&F.F3F0F0&F26#\"\"$F0F0F0%\"DGF0F0*&F7F0%\"AGF0F0
*(F7\"\"\"&F26#\"\"(F0%\"CGF0F0F?,(*&FCF?%\"BGF0F0*&&F2F&F0F=F?F0*&&F2
6#\"\"'F0F=F?F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ali
as( S=S(t), P=P(t) ): # s, p as functions of time t" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "sequence := seq( k.i = k[i], i=1..6 )," }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "            seq( k.i.i = k[i,i], i
=1..6 )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "            seq( C.i = \+
C[i], i=1..6 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(eval@su
bs)( s=sequence, 'macro(s)' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "RS := k22*C2 + (k33+k4)*C3 - (k2*E+k3*C2+k44*C1)*S;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RSG,(*&&%\"kG6$\"\"#F*\"\"\"&%\"CG6
#F*F+F+*&,&&F(6$\"\"$F3F+&F(6#\"\"%F+F+&F-6#F3F+F+*&,(*&&F(F.F+%\"EGF+
F+*&&F(F8F+F,\"\"\"F+*&&F(6$F6F6F+&F-6#F+F+F+F+%\"SGF+!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sys := " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "\{ 0 = (k11+k5)*C1 + k22*C2 - (k1+k2*S+k55*P)*E," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "  0 = (k1+k55*P)*E + k4*C3 - (k11+k
5+k44*S)*C1, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "  0 = k2*E*S + (k6
+k33)*C3 - (k22+k3*S+k66*P)*C2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "
  0 = (k66*P+k3*S)*C2 + k44*C1*S - (k33+k4+k6)*C3," }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "  E0 = C1+C2+C3+E \};  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$sysG<'/\"\"!,(*&,&&%\"kG6$\"\"\"F.F.&F,6#\"\"&F.F.&%
\"CG6#F.F.F.*&&F,6$\"\"#F8F.&F36#F8F.F.*&,(&F,F4F.*&&F,F:F.%\"SGF.F.*&
&F,6$F1F1F.%\"PGF.F.F.%\"EGF.!\"\"/F',(*&,&F=F.FAF.F.FE\"\"\"F.*&&F,6#
\"\"%F.&F36#\"\"$F.F.*&,(F+F.F/F.*&&F,6$FOFOF.F@FKF.F.F2FKFF/F',(*&,&*
&&F,6$\"\"'FinF.FDFKF.*&&F,FQF.F@FKF.F.F9FKF.*(FVFKF2FKF@FKF.*&,(&F,6$
FRFRF.FMF.&F,6#FinF.F.FPFKFF/%#E0G,*F2F.F9F.FPF.FEF./F',(*(F?FKFEFKF@F
KF.*&,&FaoF.F_oF.F.FPFKF.*&,(F6F.FjnF.FfnF.F.F9FKFF" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "solve( sys, \{C1,C2,C3,E\} ):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "k55 := 0: k66 := 0:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "diff(S,t) = collect( normal(RS), S, factor );  " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$%\"SG%\"tG*&*(F'\"\"\"%#E0G
\"\"\",(*,&%\"kG6#\"\"$F-)F'\"\"#F+&F16$\"\"%F8F-&F16#\"\"'F-&F16#F5F-
!\"\"*(F0F+,,*(&F16#F-F-F6F+F9F+F-*(&F16$F-F-F-&F16#F8F-F<F+F-*(&F16#
\"\"&F-FGF+F<F+F-*(FEF+F9F+F<F+F-*(FJF+F9F+F<F+F-F-F'F-F>**F6F+FBF+&F1
6$F3F3F-&F16$F5F5F-F-F-F+,***F<F+)F'F3F+F6F+F0F+F-*&,.*(FBF+F6F+F0F+F-
*(F<F+FGF+F0F+F-*(F<F+F6F+F9F+F-*(FJF+F0F+F<F+F-*(FEF+F0F+F<F+F-*(F<F+
F6F+FPF+F-F-F4F+F-*&,>*(FBF+F6F+FRF+F-*(FRF+F6F+F9F+F-*(FJF+FPF+F<F+F-
*(FBF+F6F+FPF+F-FAF-*(FEF+FPF+F<F+F-*(FJF+FGF+F0F+F-FIF-*(FRF+F6F+FPF+
F-FNF-*(FBF+FGF+F0F+F-*(FEF+FGF+F0F+F-FDF-FMF-F-F'F+F-*(FRF+,(FBF-FJF-
FEF-F-,(FPF-FGF-F9F-F-F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "eqns := \{ x^2 + y^2 = 25, y = x^2 - 5 \};  " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%eqnsG<$/,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF+F,F-
\"#D/F0,&F(F-!\"&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "vars
 := \{x,y\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sols := sol
ve( eqns, vars );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG6&<$/%
\"yG!\"&/%\"xG\"\"!F&<$/F+\"\"$/F(\"\"%<$/F+!\"$F0" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "map( subs, \{sols\}, eqns ); " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<$<$/!\"&F&/\"#DF(<$/\"\"%F+F'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "map2( ''eval'', eqns, \{sols\} );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%-%%evalG6$<$/,&*$)%\"xG\"\"#\"\"\"\"
\"\"*$)%\"yGF-F.F/\"#D/F2,&F*F/!\"&F/<$/F2F6/F,\"\"!-F%6$F'<$/F,\"\"$/
F2\"\"%-F%6$F'<$/F,!\"$F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 
"%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$<$/!\"&F&/\"#DF(<$/\"\"%F+F'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqns := \{ x^2+y^2=1, sqrt(x+y)=x^2
-y^2 \};  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<$/*$-%%sqrtG6#,
&%\"xG\"\"\"%\"yGF-\"\"\",&*$)F,\"\"#F/F-*$)F.F3F/!\"\"/,&F1F-F4F-F-" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "vars := \{x,y\}:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sols := solve( eqns, vars );
  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG6&<$/%\"xG,$-%'RootOfG6#
,&*$)%#_ZG\"\"#\"\"\"F1!\"\"\"\"\"F3/%\"yGF*<$/F(F4/F6\"\"!<$/F(,**$)-
F+6$,,*$)F0\"\"%F2FE*$)F0\"\"$F2FEF.!\"#F0!\"%F3F4$!+/$G*pK!#5FHF2FJF1
F4F@FH*$)F@F1F2FI/F6F@<$/F6-F+6$FB,&$!+W*p,4)FMF4%\"IG$\"+*)ovIRFM/F(,
**$)FSFHF2FJF1F4FSFH*$)FSF1F2FI" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "eval( eqns, sols[2] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#
/\"\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify( eval
( eqns, sols[1] ) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/\"\"!F%/\"
\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "map( testeq, eval
( eqns, sols[1] ) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%%trueG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert( %, `and` ); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "sols[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG,$
-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F.!\"\"\"\"\"F0/%\"yGF'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "s1 := allvalues( sols[1], 'dependen
t' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G6$<$/%\"xG,$*$-%%sqrtG6
#\"\"#\"\"\"#!\"\"F./%\"yG,$F*#\"\"\"F.<$/F(F4/F3F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "sols[1];  " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<$/%\"xG,$-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F.!\"\"\"\"\"F0/%\"yG
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "candidates := allvalu
es( sols[1], 'independent' );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+ca
ndidatesG6&<$/%\"xG,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F./%\"yG,$F*#\"\"\"F
.<$F'/F3F)<$F2/F(F4<$F:F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "for c in candidates do  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  s
implify( subs( c, eqns ) )  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "od; \+
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/\"\"!F%/\"\"\"F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<$/*$-%%sqrtG6#,$*$-F'6#\"\"#\"\"\"!\"\"F.\"\"!/
\"\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/*$)\"\"##\"\"\"\"\"%\"
\"\"\"\"!/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/\"\"!F%/\"\"\"F'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "issol := proc( sol, eqn
s )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "  convert( map( testeq, subs
( sol, eqns ) )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "    `and` )" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "end: # the selection routine" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "select( issol, \{candidates
\}, eqns );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$<$/%\"xG,$*$-%%sqrtG6
#\"\"#\"\"\"#\"\"\"F,/%\"yG,$F(#!\"\"F,<$/F&F2/F1F'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "sols[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#<$/%\"xG,**$)-%'RootOfG6$,,*$)%#_ZG\"\"%\"\"\"F0*$)F/\"\"$F1F0*$)F/
\"\"#F1!\"#F/!\"%!\"\"\"\"\"$!+/$G*pK!#5F4F1F9F7F;F)F4*$)F)F7F1F8/%\"y
GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert( %, radical \+
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG,,*$),(#!\"\"\"\"%\"\"\"
*$-%%sqrtG6#\"\"&\"\"\"#F-F,*$-F06#,&\"\"#F-F.F9F3F*\"\"$F3!\"%#F2F,F-
F.#F:F,F5#!\"$F,*$)F)F9F3!\"#/%\"yGF)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "s3 := simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#s3G<$/%\"yG,(#!\"\"\"\"%\"\"\"*$-%%sqrtG6#\"\"&\"\"\"#F,F+*$-F/6#,
&\"\"#F,F-F8F2F)/%\"xG,(F)F,F-F3F4F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "convert( sols[4], radical );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"xG,,*$),(#!\"\"\"\"%\"\"\"*$-%%sqrtG6#\"\"&\"\"\"
F**$-F06#,&\"\"#F-F.!\"#F3#F-F,\"\"$F3!\"%#F2F,F-F.#!\"$F,F4#F;F,*$)F)
F8F3F9/%\"yGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s4 := sim
plify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s4G<$/%\"yG,(#!\"\"\"
\"%\"\"\"*$-%%sqrtG6#\"\"&\"\"\"F)*&%\"IGF,-F/6#,&!\"#F,F-\"\"#F2#F,F+
/%\"xG,(F)F,F-F)F3F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s1,
 sols[2], s3, s4;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'<$/%\"xG,$*$-%%sq
rtG6#\"\"#\"\"\"#!\"\"F+/%\"yG,$F'#\"\"\"F+<$/F%F1/F0F&<$/F%F3/F0\"\"!
<$/F0,(#F.\"\"%F3*$-F)6#\"\"&F,#F3F?*$-F)6#,&F+F3F@F+F,F>/F%,(F>F3F@FD
FEFD<$/F0,(F>F3F@F>*&%\"IGF3-F)6#,&!\"#F3F@F+F,FD/F%,(F>F3F@F>FNF>" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "readlib(forget):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "forget( solve ):  # forget p
revious results" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "_EnvExpl
icit := true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "candidates
 := \{ solve( eqns ) \};  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+candi
datesG<)<$/%\"xG\"\"\"/%\"yG\"\"!<$/F+,(#!\"\"\"\"%F)*$-%%sqrtG6#\"\"&
\"\"\"#F)F2*$-F56#,&\"\"#F)F3F>F8F9/F(,,*$)F/\"\"$F8!\"%#F7F2F)F3#FCF2
F:FF*$)F/F>F8!\"#<$/F+,(F0F)F3F0*$-F56#,&F>F)F3FIF8F9/F(,,*$)FLFCF8FDF
EF)F3#!\"$F2FMFF*$)FLF>F8FI<$/F(,,*$),(F0F)F3F0FMF0FCF8FDFEF)F3FUFMFU*
$)FhnF>F8FI/F+Fhn<$/F+,(F0F)F3F9F:F0/F(,,*$)F^oFCF8FDFEF)F3FFF:FU*$)F^
oF>F8FI<$/F(,$*$-F56#F>F8#F)F>/F+,$Fho#F1F><$/F(F]p/F+Fgo" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "iss
ol := proc( sol, eqns )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  local \+
eqs;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  eqs := subs( sol, eqns );
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  eqs := map( lhs - rhs, eqs );
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "  eqs := map( u -> u^2, eqs ); \+
 # <-- square expressions" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "  eqs \+
:= radnormal( eqs );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "  if not co
nvert( map( testeq, eqs ), `and` )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "  then false" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  else true" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "  fi" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "end: # the selection routine" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "select( issol, candidates, eqns );" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#<'<$/%\"xG\"\"\"/%\"yG\"\"!<$/F),(#!\"\"\"\"%F'*$-%%s
qrtG6#\"\"&\"\"\"F.*$-F36#,&\"\"#F'F1!\"#F6#F'F0/F&,,*$)F-\"\"$F6!\"%#
F5F0F'F1#!\"$F0F7#FBF0*$)F-F;F6F<<$/F),(F.F'F1F=*$-F36#,&F;F'F1F;F6F./
F&,,*$)FLFBF6FCFDF'F1FGFMFE*$)FLF;F6F<<$/F&,$*$-F36#F;F6#F'F;/F),$FZ#F
/F;<$/F&Fin/F)FY" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "P[1] := ( lhs - rhs )( eqns[2] ); " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"\",(*$)%\"xG\"\"#\"\"\"F'*$)%\"yGF,F-
F'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P[2] := ( lhs^
2 - rhs^2 )( eqns[1] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#
\"\"#,(%\"xG\"\"\"%\"yGF**$),&*$)F)F'\"\"\"F**$)F+F'F1!\"\"F'F1F4" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Peqns := \{ P[1], P[2] \};" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PeqnsG<$,(%\"xG\"\"\"%\"yGF(*$),&
*$)F'\"\"#\"\"\"F(*$)F)F/F0!\"\"F/F0F3,(F-F(F1F(F3F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "eqn_x := resultant( P[1], P[2], y ) = 0; \+
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn_xG/,0*$)%\"xG\"\"#\"\"\"!
\"'F)!\"#*$)F)\"\"&F+!\")*$)F)\"\"$F+\"\")*$)F)\"\"%F+\"#C*$)F)F5F+\"#
;*$)F)\"\"'F+!#K\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq
n_x := factor( eqn_x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn_xG/,
$**%\"xG\"\"\",&F(F)!\"\"F)F),&*$)F(\"\"#\"\"\"F/F+F)F),,*$)F(\"\"%F0F
4*$)F(\"\"$F0F4F-!\"#F(!\"%F+F)F)F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "x_roots := solve( eqn_x );  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(x_rootsG6*\"\"!\"\"\",$*$-%%sqrtG6#\"\"#\"\"\"#F'F-,
$F)#!\"\"F-,(#F2\"\"%F'*$-F+6#\"\"&F.#F'F5*$-F+6#,&F-F'F6F-F.F:,(F4F'F
6F:F;F4,(F4F'F6F4*$-F+6#,&F-F'F6!\"#F.F:,(F4F'F6F4FAF4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify( P[2], \{ P[1] \}, [ y, x \+
] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%\"xG\"\"\"%\"yGF%*$)F$\"\"%
\"\"\"!\"%*$)F$\"\"#F*F)!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "eqn_y := y = solve(%,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&eqn_yG/%\"yG,*%\"xG!\"\"*$)F(\"\"%\"\"\"F,*$)F(\"\"#F-!\"%\"\"\"F2" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eliminate( \{P[1],P[2]\}, \+
y );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<#/%\"yG,*%\"xG!\"\"*$)F(\"
\"%\"\"\"F,*$)F(\"\"#F-!\"%\"\"\"F2<#**F(F2,&F(F2F)F2F2,&F.F0F)F2F2,,F
*F,*$)F(\"\"$F-F,F.!\"#F(F1F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "eqns := \{ x^2 + y^2 = 1,  z = x^2 - y^2,  z^2 = x + \+
y \};  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/%\"zG,&*$)%\"xG
\"\"#\"\"\"\"\"\"*$)%\"yGF,F-!\"\"/*$)F'F,F-,&F+F.F1F./,&F)F.F/F.F." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "polys := map( lhs - rhs, e
qns );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polysG<%,(*$)%\"xG\"\"#
\"\"\"\"\"\"*$)%\"yGF*F+F,!\"\"F,,(%\"zGF,F'F0F-F,,(*$)F2F*F+F,F)F0F/F
0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eliminate( polys, \{y,
z\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$<$/%\"zG,2*$)%\"xG\"\"#\"
\"\"\"\")*$)F*\"\"&F,F-*$)F*\"\"$F,!\")F*F+*$)F*F-F,!#;*$)F*\"\"'F,\"#
K*$)F*\"\"%F,!#C!\"\"\"\"\"/%\"yG,*F*F@F<F>F(!\"%FAFA<#**F*FA,&F*FAF@F
AFA,&F(F+F@FAFA,,F<F>F1F>F(!\"#F*FEF@FAFA" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "Groebner" {TEXT -1 30 "1
6.5 The Groebner Basis Method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "polys := \+
\{ x^2 + y^2 - 1, x^2 - y^2 - z,  x + y - z^2 \};  " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&polysG<%,(%\"xG\"\"\"%\"yGF(*$)%\"zG\"\"#\"\"\"!\"
\",(*$)F'F-F.F(*$)F)F-F.F/F,F/,(F1F(F3F(F/F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "with( Groebner ):  # load the Groebner basis pac
kage" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := gbasis( polys,
 plex(z,y,x) );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%,0*$)%\"x
G\"\"#\"\"\"!\"$*$)F)\"\"&F+!\"%*$)F)\"\")F+F3*$)F)\"\"'F+!#;*$)F)\"\"
%F+\"#7*$)F)\"\"$F+F:F)!\"\",,F)\"\"\"%\"yGFAF8F0F'F:F?FA,(%\"zGFAF'!
\"#FAFA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "G := factor( G )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%**%\"xG\"\"\",&F'F(!\"\"F
(F(,&*$)F'\"\"#\"\"\"F.F*F(F(,,*$)F'\"\"%F/F3*$)F'\"\"$F/F3F,!\"#F'!\"
%F*F(F(,,F'F(%\"yGF(F1F8F,F3F*F(,(%\"zGF(F,F7F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "_EnvExplicit := true:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "assign( \{ solve(G[3],\{z\}), solve(G[2],\{y\}
) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rootlist := [ so
lve( G[1] ) ];  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)rootlistG7*\"\"
!\"\"\",$*$-%%sqrtG6#\"\"#\"\"\"#F'F-,$F)#!\"\"F-,(#F2\"\"%F'*$-F+6#\"
\"&F.#F'F5*$-F+6#,&F-F'F6F-F.F:,(F4F'F6F:F;F4,(F4F'F6F4*$-F+6#,&F-F'F6
!\"#F.F:,(F4F'F6F4FAF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f
or x in rootlist do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "  simplify(
 [ 'x'=x, 'y'=y, 'z'=z ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "  if
 simplify( subs( %, polys ) ) = \{0\}" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "  then print( `valid solution` )" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "  else print( `INvalid solution` ) " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "  fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "od;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"xG\"\"!/%\"yG\"\"\"/%\"zG!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~solutionG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7%/%\"xG\"\"\"/%\"yG\"\"!/%\"zGF&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%/valid~solutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7%/%\"xG,$*$-%%sqrtG6#\"\"#\"\"\"#\"\"\"F+/%\"yG,$F'#!\"\"F+/%\"zG\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~solutionG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%/%\"xG,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F+/%\"yG,$F
'#\"\"\"F+/%\"zG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~solut
ionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"xG,(#!\"\"\"\"%\"\"\"*$-
%%sqrtG6#\"\"&\"\"\"#F*F)*$-F-6#,&\"\"#F*F+F6F0F1/%\"yG,(F'F*F+F1F2F'/
%\"zG,&F2F'*&F,F0F3F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~sol
utionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"xG,(#!\"\"\"\"%\"\"\"*
$-%%sqrtG6#\"\"&\"\"\"#F*F)*$-F-6#,&\"\"#F*F+F6F0F'/%\"yG,(F'F*F+F1F2F
1/%\"zG,&F2F1*&F,F0F3F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~s
olutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"xG,(#!\"\"\"\"%\"\"
\"*$-%%sqrtG6#\"\"&\"\"\"F'*&%\"IGF*-F-6#,&!\"#F*F+\"\"#F0#F*F)/%\"yG,
(F'F*F+F'F1F'/%\"zG,&F1F'*(F2F0F,F0F3F0F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%/valid~solutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%
/%\"xG,(#!\"\"\"\"%\"\"\"*$-%%sqrtG6#\"\"&\"\"\"F'*&%\"IGF*-F-6#,&!\"#
F*F+\"\"#F0F'/%\"yG,(F'F*F+F'F1#F*F)/%\"zG,&F1F;*(F2F0F,F0F3F0F;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%/valid~solutionG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "eqns := [ c*x + x*y^2 + x*z^2 = 1," }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 55 "  c*y + y*x^2 + y*z^2 = 1, c*z + z*x^2 + z*y^2 = 1
 ];  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG7%/,(*&%\"cG\"\"\"%\"
xGF*F**&F+\"\"\")%\"yG\"\"#F-F**&F+F-)%\"zGF0F-F*F*/,(*&F)F-F/F*F**&F/
F-)F+F0F-F**&F/F-F2F-F*F*/,(*&F)F-F3F*F**&F3F-F8F-F**&F3F-F.F-F*F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "_EnvExplicit := false: # st
ick to RootOfs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve( \{
op(eqns)\}, \{x,y,z\} );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6'<%/%\"yG
-%'RootOfG6#,(*$)%#_ZG\"\"$\"\"\"\"\"#*&%\"cG\"\"\"F,F2F2!\"\"F2/%\"zG
F&/%\"xG*&F.F.,&F1F2*$)F&F/F.F/!\"\"<%/F5-F'6#,**$)F,\"\"%F.F/*&F1F.)F
,F/F.F-*$)F1F/F.F2F,F3/F%,$*&,(F3F2*&F1F.F?F2F/*$)F?F-F.F/F.F1F<F3/F7,
$*&*$FHF.F.,(F3F2FMF-FNF/F<F3<%/F%F?/F5FJFP<%F>/F%,$*&,(*&F1F.)F?F/F.F
2FGF2F?F2F.,(FNF2FMF2F3F2F<#F3F//F7,$*&,,!\"%F2FN\"\"'*&FHF.FhnF.F2FM
\"\"(*$)F1F-F.F2F.,**&F1F.FOF.F2*&F?F.FHF.F2F1!\"#*$FhnF.F3F<#F2F/<%/F
%-F'6#,*F1F2*&-F'6#,(F2F2F*F2F0F2F2F,F.F2*$FFF.F2*$)FapF/F.F2/F7*&,&F1
F2FepF2F.F]pF</F5Fap" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pol
ys := map( lhs - rhs, eqns );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
polysG7%,**&%\"cG\"\"\"%\"xGF)F)*&F*\"\"\")%\"yG\"\"#F,F)*&F*F,)%\"zGF
/F,F)!\"\"F),**&F(F,F.F)F)*&F.F,)F*F/F,F)*&F.F,F1F,F)F3F),**&F(F,F2F)F
)*&F2F,F7F,F)*&F2F,F-F,F)F3F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "with( Groebner ):  # load Groebner basis package" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "settime := time():" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sys := gsolve( polys, \{z,y,x,c\} )
;  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<'7%7%,&%\"yG\"\"\"%\"zG
!\"\",&%\"xGF*F+F,,(*&%\"cGF*F+F*F**$)F+\"\"$\"\"\"\"\"#F,F*-%%plexG6&
F1F.F)F+<#,&F,F*F2F67%7%F(,(*&F+F5)F.F6F5F**&F.F*)F+F6F5F*F,F*,(F1F**&
F.F5F+F5F,*$FBF5F*F7<\"7%7%,(*&F+F5)F)F6F5F**&F)F*FBF5F*F,F*,&F.F*F)F,
,(F1F**$FKF5F**&F)F5F+F5F,F7<$F(,&F,F**$)F)F4F5F67%7%FI,(F.F*F)F*F+F*,
*F1F*FOF*FPF*FEF*F7<#F(7%7%FIF-,(F1F*FPF,FEF*F7<$F(F;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "cpu_time := (time()-settime) * seco
nds; # computing time" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)cpu_timeG,
$%(secondsG$\"%ff!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fo
r s in sys do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "    solve( \{op(s
[1])\}, \{x,y,z\} ) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG-%'RootOfG6#,(*&%\"cG\"\"\"%#_
ZGF,F,*$)F-\"\"$\"\"\"\"\"#!\"\"F,/%\"zGF&/%\"xGF&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<%/%\"yG-%'RootOfG6#,**$)%\"cG\"\"#\"\"\"\"\"\"*&F,F/
)%#_ZGF-F.\"\"$*$)F2\"\"%F.F-F2!\"\"/%\"zGF&/%\"xG,$*&,(F7F/*&F,F.F&F/
F-*$)F&F3F.F-F.F,!\"\"F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG-%
'RootOfG6#,**$)%\"cG\"\"#\"\"\"\"\"\"*&F,F/)%#_ZGF-F.\"\"$*$)F2\"\"%F.
F-F2!\"\"/%\"xGF&/%\"zG,$*&,(F7F/*&F,F.F&F/F-*$)F&F3F.F-F.F,!\"\"F7" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG-%'RootOfG6#,(\"\"\"F**&%\"cG
F*%#_ZGF*F**$)F-\"\"$\"\"\"F*/%\"zG-F'6#,*F,F**$)F&\"\"#F1F**&F&F*F-F1
F**$)F-F9F1F*/%\"xG,&F&!\"\"F4F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%
/%\"xG-%'RootOfG6#,**$)%\"cG\"\"#\"\"\"\"\"\"*&F,F/)%#_ZGF-F.\"\"$*$)F
2\"\"%F.F-F2!\"\"/%\"zGF&/%\"yG,$*&,(F7F/*&F,F.F&F/F-*$)F&F3F.F-F.F,!
\"\"F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "settime := time()
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "G := gbasis( polys, pl
ex(x,y,z) ):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, computation inte
rrupted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "cpu_time := (tim
e()-settime) * seconds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)cpu_time
G,$%(secondsG$\"$V'!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
factor( G[-1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%\"cG\"\"\"%\"
xGF&F&*&F'\"\"\")%\"yG\"\"#F)F&*&F'F))%\"zGF,F)F&!\"\"F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "sys := [ x - (N+h)*cf*cl, y - (N+h)*cf*sl," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "  z - (N*(1-e^2)+h)*sf, cf^2 + sf^2
 - 1, cl^2 +sl^2 - 1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "  tf*cf - \+
sf, N*S - a, S^2 + e^2*sf^2 - 1, (N+h)*cf - d, " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "  d^2 - x^2 - y^2 ];  " }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$sysG7,,&%\"xG\"\"\"*(,&%\"NGF(%\"hGF(F(%#cfGF(%#clGF(!\"\",&%
\"yGF(*(F*F(F-F(%#slGF(F/,&%\"zGF(*&,&*&F+F(,&F(F(*$)%\"eG\"\"#\"\"\"F
/F(F(F,F(F(%#sfGF(F/,(*$)F-F=F>F(*$)F?F=F>F(F/F(,(*$)F.F=F>F(*$)F3F=F>
F(F/F(,&*&%#tfGF(F-F(F(F?F/,&*&F+F(%\"SGF(F(%\"aGF/,(*$)FOF=F>F(*&)F<F
=F>)F?F=F>F(F/F(,&*&F*F(F-F(F(%\"dGF/,(*$)FYF=F>F(*$)F'F=F>F/*$)F1F=F>
F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with( Groebner ):  # \+
load Groebner basis package" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "vars := N,S,x,y,h,cl,sl,cf,sf,tf: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "gsys := gbasis( sys, plex(vars) ):  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "collect( gsys[1], tf );  # get the \+
polynomial in tf  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&*&)%\"dG\"
\"#\"\"\")%\"eGF)F*\"\"\"*$)F(F)F*!\"\"F-)%#tfG\"\"%F*F-*&,&*()F,F)F*F
(F-%\"zGF-!\"#*&F(F-F8F-F)F-)F2\"\"$F*F-*&,**&)F,F3F*)%\"aGF)F*F-*&)F8
F)F*)F,F)F*F-*$)F8F)F*F0F.F0F-)F2F)F*F-*(F(F-F8F-F2F-F)FFF0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "map( convert, - %, sqrfree )
; # rewrite the polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*()%#t
fG\"\"%\"\"\")%\"dG\"\"#F(,&!\"\"\"\"\"*$)%\"eGF+F(F.F.F-**)F&\"\"$F(F
*F.%\"zGF.F,F.F+*&,**&)F1F'F()%\"aGF+F(F.*&)F5F+F()F1F+F(F.*$)F5F+F(F-
*$)F*F+F(F-F.)F&F+F(F-*(F*F.F5F.F&F.!\"#F?F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "sort( subs( e^2*z^2 = (e^2-1)*z^2 + z^2, % ), tf
 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*()%\"dG\"\"#\"\"\",&!\"\"
\"\"\"*$)%\"eGF'F(F+F+)%#tfG\"\"%F(F***F&F+%\"zGF+F)F+)F0\"\"$F(F'*&,(
*&)F.F1F()%\"aGF'F(F+*&F)F+)F3F'F(F+*$)F&F'F(F*F+)F0F'F(F**(F&F+F3F+F0
F+!\"#*$)F3F'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "univpo
ly(tf,sys,\{vars\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*()%#tfG\"
\"#\"\"\")%\"aGF'F()%\"eG\"\"%F(\"\"\"*()F,F'F()F&F'F()%\"zGF'F(F.**)F
,F'F()F&\"\"$F(%\"dGF.F3F.!\"#*()F,F'F()F&F-F()F8F'F(F.*&)F&F'F()F3F'F
(!\"\"*$)F3F'F(FA*(F8F.F3F.F&F.F'*()F&F7F(F8F.F3F.F'*&)F8F'F()F&F'F(FA
*&)F&F-F()F8F'F(FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "colle
ct(%, tf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&*&)%\"dG\"\"#\"\"
\")%\"eGF)F*\"\"\"*$)F(F)F*!\"\"F-)%#tfG\"\"%F*F-*&,&*&F(F-%\"zGF-F)*(
)F,F)F*F(F-F7F-!\"#F-)F2\"\"$F*F-*&,**&)F,F3F*)%\"aGF)F*F-*&)F7F)F*)F,
F)F*F-*$)F7F)F*F0F.F0F-)F2F)F*F-*(F(F-F7F-F2F-F)FFF0" }}}}{SECT 1 
{PARA 3 "" 0 "Inequalities" {TEXT -1 17 "16.6 Inequalities" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "plot( x^3 + 4*x^2 + 2*x - 1, x = -4..1 );" }}{PARA 
13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7W7$$!\"%\"\"!$!\"*F*7$$!1LLe9r]XR!
#:$!1j7vV?sU!)F07$$!1nm;HU,\"*QF0$!1wq0*y%)>8(F07$$!1+v$4E+O%QF0$!1b)4
*[gmwjF07$$!1L$3FH'='z$F0$!1b)G+12_l&F07$$!1+DcEV'Gu$F0$!19\"zSU1N)[F0
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0$!11Y,Z>cFk!#;7$$!1+]PM.ttKF0$\"13GL3q'>O#Fjn7$$!1n;/,olnJF0$\"1hF8[$
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0$\"1A^:Xy(oS$F07$$!1LL$eaR%HBF0$\"1HM`_S11MF07$$!1LLL$o#)RB#F0$\"1L1$
[^fcM$F07$$!1+]PfO%H7#F0$\"1*y\\4s)z8KF07$$!1LLL3`lC?F0$\"1Xbjg)z![IF0
7$$!1+]P4u\"o\">F0$\"1b=*3qr.#GF07$$!1+]7G-89=F0$\"1\\2ShmelDF07$$!1LL
3Fp)pq\"F0$\"1BsScwTnAF07$$!1L$3-$ff3;F0$\"1hO3b&f2(>F07$$!1nm\"z%zY-:
F0$\"16SJqx,L;F07$$!1m;/,3B#R\"F0$\"10-KcY>q7F07$$!1+]iS!piH\"F0$\"1e?
A#\\td]*Fjn7$$!1mmT5xi#>\"F0$\"1M!p6J5%ygFjn7$$!1+++vgb&3\"F0$\"1I)3-5
=Oj#Fjn7$$!1****\\(ot!3)*Fjn$!1ZAQ\"pY-s&!#<7$$!1(**\\P4wXz)Fjn$!1OJ&f
m^MX$Fjn7$$!1++]7)f#pwFjn$!1*)*=*>;PAjFjn7$$!1PLL$38\"emFjn$!1&[ph/6c`
)Fjn7$$!1)****\\7)\\ybFjn$!1V%)eK[^W5F07$$!1pmT5l?+YFjn$!1>RT9W\"4<\"F
07$$!1%****\\Pv1`$Fjn$!1'\\E+c?:D\"F07$$!1NL3F*HV_#Fjn$!1k3Ji>1m7F07$$
!1'**\\(o*3CZ\"Fjn$!1#GF@Va4@\"F07$$!1bLL$eR'RWF^v$!12^))Qh*43\"F07$$
\"1x**\\ilAFjF^v$!1*QMYZ')=d)Fjn7$$\"1QLLL)*pp;Fjn$!1L2\\*z\"*))\\&Fjn
7$$\"1SL$3xe,t#Fjn$!1K6f@gna8Fjn7$$\"1tm\"HdO=y$Fjn$\"1HuuWuZDQFjn7$$
\"10++]#>#[ZFjn$\"1&4$4vn=&e*Fjn7$$\"1um;aG!e&eFjn$\"1\\n>McdV<F07$$\"
1RLLL)Qk%oFjn$\"1e*[O%[:lDF07$$\"1,+D1Mm-zFjn$\"1l=WvS:sNF07$$\"11+v$4
0O\"*)Fjn$\"1<6_s@-pYF07$$\"\"\"F*$\"\"'F*-%'COLOURG6&%$RGBG$\"#5!\"\"
F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fj[l%(DEFAULTG" 2 400 300 
300 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 1 0 0 0 0 4192 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 38 "solve( x^3 + 4*x^2 + 2*x - 1 > 0, x );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%OpenG6#,&#!\"$\"\"#\"\"\"*
$-%%sqrtG6#\"#8\"\"\"#!\"\"F,-F'6#F5-F$6$-F'6#,&F*F-F.#F-F,%)infinityG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve( \{ x^3 + 4*x^2 +
 2*x - 1 > 0 \}, \{x\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$2,&#!\"
$\"\"#\"\"\"*$-%%sqrtG6#\"#8\"\"\"#!\"\"F(%\"xG2F2F1<#2,&F&F)F*#F)F(F2
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve( abs(x)*x + exp(x
)>=0, \{x\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#1,$-%)LambertWG6##
\"\"\"\"\"#!\"#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sol
ve( a*x + b >= c, \{x\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#1,$*&-
%'signumG6#%\"aG\"\"\"%\"xGF+!\"\"*&*&F'\"\"\",&%\"cGF-%\"bGF+F+F0F*!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solve( \{ x^2 - x =
 0, x<>0 \}, x ); # nonzero solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#<#/%\"xG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "_EnvExp
licit := true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve( \{
x^4-9, x^2-3<>0\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/%\"xG*&%\"I
G\"\"\"-%%sqrtG6#\"\"$\"\"\"<#/F%,$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "solve( \{ x<1/3, y<1/2, x^2+y^2=1 \}, \{x,y\} );" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf([%]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7$<#/%\"xG,$%\"IG$\"+330K<!\"*<#/F&,$F($!+330K<F
+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "Numerical" {TEXT -1 22 "16.7 Numerical Solvers" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "x^7 - 2*x^6 - 4*x^5 - x^3 + x^2 + 6*x + 4;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG\"\"(\"\"\"\"\"\"*$)F&\"\"'F
(!\"#*$)F&\"\"&F(!\"%*$)F&\"\"$F(!\"\"*$)F&\"\"#F(F)F&F,\"\"%F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fsolve(%);  " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%$!+xz1O7!\"*$\"+yRIn6F%$\"+xz1OKF%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve( %% , x , complex ); " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6)$!+xz1O7!\"*,&$!+OV%)[w!#5\"\"\"%\"IG$
!+garCNF),&F'F*F+$\"+garCNF),&$\"+XWK7=F)F*F+$!+,T&R3\"F%,&F2F*F+$\"+,
T&R3\"F%$\"+yRIn6F%$\"+xz1OKF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "fsolve( %%% , x , 0..2 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+yRIn6!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn := si
n(x) = x/2;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$sinG6#%\"x
G,$F)#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve
( eqn, x );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve( eqn, x, avoid = \{x=0\} );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+nU\\&*=!\"*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "fsolve( eqn, x, avoid = \{x=0, x=%\} );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nU\\&*=!\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "fsolve( eqn, x, avoid = \{x=0, x=%%, x=%\} );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6%/-%$sinG6#%\"xG,$F*#\"
\"\"\"\"#F*/%&avoidG<%/F*\"\"!/F*$\"+nU\\&*=!\"*/F*$!+nU\\&*=F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "sols := \{\}:\ndo \n  newso
l := fsolve( eqn, x, avoid = sols ):\n  if type(newsol, 'function') th
en \n    break \n  else\n    sols := \{op(sols), x=newsol\}\n  fi\nod:
\nsols;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG$!+nU\\&*=!\"*/F%\"
\"!/F%$\"+nU\\&*=F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fsol
ve( eqn, x, 0.1 .. infinity );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+nU\\&*=!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve( e
qn, x, -0.1 .. 0.1 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fsolve( eqn, x, -infinity ..
 -0.1 ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+nU\\&*=!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(sin(x),x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "fsolve(sin(x),x=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEf
TJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "readlib( realroot
 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x^7 - 2*x^6 - 4*x^5 \+
- x^3 + x^2 + 6*x + 4;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG
\"\"(\"\"\"\"\"\"*$)F&\"\"'F(!\"#*$)F&\"\"&F(!\"%*$)F&\"\"$F(!\"\"*$)F
&\"\"#F(F)F&F,\"\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "re
alroot(%);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$\"\"!\"\"#7$F&\"\"
%7$!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "realroot( %
%, 1/100 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$#\"$\\\"\"$G\"#\"
#v\"#k7$#\"$2#F*#\"$:%F'7$#!$f\"F'#!#zF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "readlib(sturm):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "x^7 - 2*x^6 - 4*x^5 - x^3 + x^2 + 6*x + 4;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG\"\"(\"\"\"\"\"\"*$)F&\"\"'F
(!\"#*$)F&\"\"&F(!\"%*$)F&\"\"$F(!\"\"*$)F&\"\"#F(F)F&F,\"\"%F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sturm( %, x, -infinity,infin
ity ); # three real roots" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sturm( %%, x, 0, infinity )
;  # two positive roots" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sturm( %%%, x, 2, 4 );  # on
e root between 2 and 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "O
thers" {TEXT -1 27 "16.8 Other Solvers in Maple" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 49 "nondimensional := force^f * speed^v * diameter^d " }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "  * density^r * acoustic_velocity^c " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  * kinematic_viscosity^m;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/nondimensionalG*.)%&forceG%\"fG\"\"
\")%&speedG%\"vGF))%)diameterG%\"dGF))%(densityG%\"rGF))%2acoustic_vel
ocityG%\"cGF))%4kinematic_viscosityG%\"mGF)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "subs( \{ force = M*L/T^2, speed = L/T, diameter = \+
L,  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "  density = M/L^3, acoustic
_velocity = L/T," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "  kinematic_vis
cosity = L^2/T \}, nondimensional );  " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*.)*&*&%\"MG\"\"\"%\"LGF(\"\"\"*$)%\"TG\"\"#F*!\"\"%\"fGF()*&F)F
*F-F/%\"vGF()F)%\"dGF()*&F'F**$)F)\"\"$F*F/%\"rGF()F2%\"cGF()*&*$)F)\"
\"#F*F*F-F/%\"mGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simpl
ify( %, 'symbolic' );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()%\"MG,&%
\"fG\"\"\"%\"rGF(F()%\"LG,.F'F(%\"vGF(%\"dGF(F)!\"$%\"cGF(%\"mG\"\"#F(
)%\"TG,*F'!\"#F-!\"\"F0F7F1F7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "eqns := \{ seq( op(2,i)=0, i= % ) \};  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqnsG<%/,&%\"fG\"\"\"%\"rGF)\"\"!/,.F(F)%\"vGF)%\"dG
F)F*!\"$%\"cGF)%\"mG\"\"#F+/,*F(!\"#F.!\"\"F1F7F2F7F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "isolve(%); # find all integral solu
tions  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(/%\"fG,$%$_N1G!\"\"/%\"vG
,(F'\"\"#%$_N2GF(%$_N3GF(/%\"dG,&F'F,F.F(/%\"rGF'/%\"cGF-/%\"mGF." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs( %, nondimensional );  \+
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*.)%&forceG,$%$_N1G!\"\"\"\"\")%&s
peedG,(F'\"\"#%$_N2GF(%$_N3GF(F))%)diameterG,&F'F-F/F(F))%(densityGF'F
))%2acoustic_velocityGF.F))%4kinematic_viscosityGF/F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify( ln(%), 'symbolic' );" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,4*&%$_N1G\"\"\"-%#lnG6#%&forceGF&!\"
\"*&F%\"\"\"-F(6#%&speedGF&\"\"#*&%$_N2GF&F.F-F+*&%$_N3GF&F.F-F+*&F%F-
-F(6#%)diameterGF&F1*&F5F-F7F-F+*&F%F--F(6#%(densityGF&F&*&F3F--F(6#%2
acoustic_velocityGF&F&*&F5F--F(6#%4kinematic_viscosityGF&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "[ coeffs( %, \{_N1,_N2,_N3\}
, 'vars' ) ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%,*-%#lnG6#%&forceG!
\"\"-F&6#%&speedG\"\"#-F&6#%(densityG\"\"\"-F&6#%)diameterGF-,(F*F)F2F
)-F&6#%4kinematic_viscosityGF1,&F*F)-F&6#%2acoustic_velocityGF1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "combine( %, ln, anything, 's
ymbolic' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%#lnG6#*&*()%&speedG
\"\"#\"\"\"%(densityG\"\"\")%)diameterGF+F,F,%&forceG!\"\"-F%6#*&%4kin
ematic_viscosityGF,*&F*\"\"\"F0\"\"\"F2-F%6#*&%2acoustic_velocityGF,F*
F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "zip( (a,b) -> exp(a)^
b, %, [vars] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%)*&*()%&speedG\"
\"#\"\"\"%(densityG\"\"\")%)diameterGF)F*F*%&forceG!\"\"%$_N1G)*&%4kin
ematic_viscosityGF**&F(\"\"\"F.\"\"\"F0%$_N3G)*&%2acoustic_velocityGF*
F(F0%$_N2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "convert( %, `
*` );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()*&*()%&speedG\"\"#\"\"\"%(
densityG\"\"\")%)diameterGF)F*F*%&forceG!\"\"%$_N1GF,)*&%4kinematic_vi
scosityGF**&F(\"\"\"F.\"\"\"F0%$_N3GF,)*&%2acoustic_velocityGF*F(F0%$_
N2GF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs( _N1=0, _N2=-
1, _N3=0, % ); # the Mach number  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*&%&speedG\"\"\"%2acoustic_velocityG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 58 "subs( _N1=0, _N2=1, _N3=-1, %% ); # the Reynold's n
umber  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%)diameterG\"\"\"%2acou
stic_velocityGF&\"\"\"%4kinematic_viscosityG!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "msolve( y^2 = x^3 - 28, 7 );  " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6)<$/%\"yG\"\"!/%\"xGF&<$/F%\"\"\"/F(\"\"#<$F*/F(
\"\"%<$F*/F(F+<$F,/F%\"\"'<$F/F4<$F2F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 
"rsolve( \{ f(n+2) = f(n+1) + f(n), " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "  f(0)=0, f(1)=1 \}, f(n) ); # Fibonacci numbers  " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*&*&,&*$-%%sqrtG6#\"\"&\"\"\"#\"\"\"F+!\"\"F.F
.),$*&F,F,,&F.F.F'F/!\"\"!\"#%\"nGF.F,F3F4F.*&*&,&F'#F/F+F/F.F.),$*&F,
F,,&F.F.F'F.F4F5F6F.F,F>F4F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "rsolve( \{ f(n+2) = x*f(n+1) + y*f(n),  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 " f(0)=0, f(1)=1 \}, f(n) );   " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&*&,(*&%\"xG\"\"\"-%%sqrtG6#,&*$)F(\"\"#\"\"\"F)%\"y
G\"\"%F1F)F.!\"\"F2!\"%F)),$*&F2F1,&F(F)*$F*F1F4!\"\"!\"#%\"nGF)F1*&F-
\"\"\"F9\"\"\"F;F)*&*&,(F.F)F'F)F2F3F)),$*&F2F1,&F(F)F:F)F;F<F=F)F1*&F
-\"\"\"FG\"\"\"F;F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "norm
al( %, 'expanded' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&)\"\"#%
\"nG\"\"\")*&%\"yG\"\"\",&%\"xG!\"\"*$-%%sqrtG6#,&*$)F/F'F-F)F,\"\"%F-
F)!\"\"F(F)F)),$*&F,F-,&F/F)F1F)F9!\"#F(F0F-*$-F36#F5F-F9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "normal( subs( n=5,% ), 'expanded' )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"\"\"\"*&%\"y
GF))F&\"\"#F(\"\"$*$)F+F-F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rsolve( \{
 T(n) = T(n-1) + n^2, T(1)=0 \}, T(n) );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*(,&%\"nG\"\"\"F'F'F',&F&#F'\"\"#F'F'F',&F&#F'\"\"$F'
F'F'F**&F%\"\"\"F(F/!\"$F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "factor(%);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"nG\"\"
\"!\"\"F'F',(*$)F&\"\"#\"\"\"F,F&\"\"&\"\"'F'F'#F'F/" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "rsolve( \{ T(n) = 2*T(n/2) + n-1, T(1)=0 \}, T(n) ); \+
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG\"\"\",()#F%\"\"#,&*&-%#ln
G6#F$\"\"\"-F-6#F)!\"\"F%F%F%F)F+F%!\"\"F%F%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "simplify(%);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,$*&,(-%#lnG6#\"\"#!\"\"*&-F'6#%\"nG\"\"\"F.F/F**&F.\"\"\"F&F/F/F1F&
!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rsolve( \{ T(n) = 3*T(n/2) +
 n, T(1)=1\}, T(n) );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)%\"nG*&-
%#lnG6#\"\"$\"\"\"-F(6#\"\"#!\"\"\"\"\"*&F$F0,&)#F.F*,&*&-F(6#F%F+F,F/
F0F0F0!\"$F.F0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simpl
ify(%);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)%\"nG*&-%#lnG6#\"\"$\"
\"\"-F(6#\"\"#!\"\"F*F%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "rsolve( a(n+1) = ln(n+1)*a(n) + 1, a(n) );   " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/-%\"aG6#\"\"!F%/-F&6#%\"nG*&-%(productG6$-%#lnG6#,&%
$_n1G\"\"\"F6F6/F5;F6,&F,F6!\"\"F6F6,(-%$sumG6$*&\"\"\"F@-F/6$F1/F5;F6
%$_n2G!\"\"/FEF8F6*&-F26#\"\"#F6F%F6F6F6F6F6" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "rsolve( u(n+1) = ln( u(n)+1 ), u(n) );  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'rsolveG6$/-%\"uG6#,&%\"nG\"\"\"F,F,
-%#lnG6#,&-F(6#F+F,F,F,F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "asympt( %, n, 4 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"\"
F%%\"nG!\"\"\"\"#*&,&%#_CG\"\"\"-%#lnG6#F&#F(\"\"$F%*$)F&\"\"#F%F'F,-%
\"OG6#*&F%F%*$)F&\"\"$F%F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "infolevel[rsolve]:= 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "rsolve( \{ x(n+2) = x(n+1) + x(n)/(n+1), " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "  x(0)=a, x(1)=b \}, x(n) );      " }}{PARA 6 "" 1 "
" {TEXT -1 85 "LREtools/resolveinits:   Warning: no solution found wit
h the given initial conditions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'r
solveG6$<%/-%\"xG6#,&%\"nG\"\"\"\"\"#F-,&-F)6#,&F,F-F-F-F-*&-F)6#F,\"
\"\"F2!\"\"F-/-F)6#\"\"!%\"aG/-F)6#F-%\"bGF4" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "readlib( ztrans ):  # load library function" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x(0) := a:  x(1) := b:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subs( ztrans(x(n),n,z) = F(z
),  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "  ztrans( (n+1)*x(n+2)=(n+1
)*x(n+1)+x(n), n, z ) );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**&%\"
zG\"\"\",**&F&\"\"\"-%\"FG6#F&F*\"\"#*&)F&F.F'-%%diffG6$F+F&F*F**&%\"a
GF*F&F'!\"#%\"bG!\"\"F*F8*&F0F'F+F'F**&F5F'F0F'F8*&F7F*F&F'F8,**&F&F',
(F+F**&F&F'F1F'F*F5F8F*F8F)F*F4F8F+F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%
\"zG\"\"#\"\"\"-%\"FG6#F'\"\"\"!\"\"*&)F'\"\"$F)-%%diffG6$F*F'F-F.*&%
\"aGF-F&F)F-,&*&F&F)F2F)F.F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "dsolve( %, F(z) );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"F
G6#%\"zG,&*(%\"aG\"\"\"F'\"\"\"-%$expG6#,&*&,(*&-%#lnG6#,&F'F+!\"\"F+F
+F'F+!\"#F8F+*&-F5F&F+F'F,F+F,F'!\"\"F+*&F,F,F'F<F+F+F+*&-F.6#F1F+%$_C
1GF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"zG,&*&*&%\"aG\"\"\")F'\"\"
#\"\"\"F/*$),&F'F,!\"\"F,\"\"#F/!\"\"F,*&*&F'F,%$_C1GF,F/*&)F2\"\"#F/-
%$expG6#*&F/F/F'F5\"\"\"F5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "subs( invztrans(F(z),z,n)=x(n), invztrans(%,z,n) );  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"nG,(*&%\"aG\"\"\"F'F+F+F*F+*&%$_
C1GF+-%*invztransG6%*&%\"zG\"\"\"*&-%$expG6#*&F3F3F2!\"\"\"\"\",(*$)F2
\"\"#F3F+F2!\"#F+F+\"\"\"F9F2F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "simplify( factor(%) );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6#%\"nG,(*&%\"aG\"\"\"F'F+F+F*F+*&%$_C1GF+-%*inv
ztransG6%*&*&%\"zGF+-%$expG6#,$*&\"\"\"F9F3!\"\"!\"\"F+F9*$),&F3F+F;F+
\"\"#F9F:F3F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := s
ubs( body = invztrans(exp(-1/z),z,k), k->body );  " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGR6#%\"kG6\"6$%)operatorG%&arrowGF(*&)!\"\"9$\"
\"\"-%*factorialG6#F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "g := subs( body = invztrans(z/(z-1)^2,z,k), k->body )
;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"kG6\"6$%)operatorG%
&arrowGF(9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "lhs(%%
%) = subsop( 3 = _C1*Sum(f(n-k)*g(k),k=0..n), " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "  rhs(%%%));  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"nG,(*&%\"aG\"\"\"F'F+F+F*F+*&%$_C1GF+-%$SumG6$*&*&)!\"\",&F'F
+%\"kGF4F+F6F+\"\"\"-%*factorialG6#F5!\"\"/F6;\"\"!F'F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(value@subs)( n=1, % );  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#\"\"\",&%\"aG\"\"#%$_C1GF'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%,_C1);  " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,&%\"bG\"\"\"%\"aG!\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "subs( _C1=%, %%% ); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6#%\"nG,(*&%\"aG\"\"\"F'F+F+F*F+*&,&%\"bGF+F*!\"
#F+-%$SumG6$*&*&)!\"\",&F'F+%\"kGF6F+F8F+\"\"\"-%*factorialG6#F7!\"\"/
F8;\"\"!F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"nG,(*&%\"aG\"\"\"F'F+F
+F*F+*&*(,&%\"bGF+F*!\"#F+)!\"\",&F'F+F2F+F+-%*hypergeomG6%7$\"\"#,&F'
F2F+F+7\"F+F+\"\"\"-%*factorialG6#F3!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 47 "match( x^2 + 2*x + 3 = (x+a)^2 + b, x, 'sol' );" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "sol;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG\"\"\"
/%\"bG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "typematch( (
x+1)^2 + 2, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "  (a::anything) &+ \+
(b::anything) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a,b;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$*$),&%\"xG\"\"\"F'F'\"\"#\"\"\"F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "typematch( (x+1)^2 + 2, " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 56 "(((u::anything) &+ (v::anything))^2) &+ (w::anythin
g) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "typematch( x^2 + 2*x + 3, " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "(((p::anything) &+ (q::anything))^2) &+ (r::anythi
ng) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "6 22 0 0" 0 }{VIEWOPTS 1 1 0 
1 1 1803 }