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{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 15" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1) " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart
;assume(x,real,y,real);expr1:=" }{TEXT -1 0 "" }{MPLTEXT 1 0 39 "cosh(
x)+sinh(y);expr2:=sinh(x)+cosh(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&expr1G,&-%%coshG6#%#x|irG\"\"\"-%%sinhG6#%#y|irGF*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&expr2G,&-%%sinhG6#%#x|irG\"\"\"-%%coshG6#%#y|ir
GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol:=solve(\{expr1=2
,expr2=5\},\{x,y\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%#x
|irG-%#lnG6#-%'RootOfG6#,(*$)%#_ZG\"\"#\"\"\"\"\"$*&\"#>F3F1F3!\"\"\"
\"(F7/%#y|irG-F)6#,&F+F7F8F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "solve(3*x^2-19*x-7=0); " }{TEXT -1 49 "aidons Maple \340 extrair
e les racines de ce trin\364me" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#
\"#>\"\"'\"\"\"*&#F'F&F'-%%sqrtG6#\"$X%F'F',&F$F'*&#F'F&F'*$F*F'F'!\"
\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Pu
isque la seconde racine est n\351gative, la solution du syst\350me est
 unique : " }{XPPEDIT 18 0 "x = ln((19+sqrt(445))/6);" "6#/%\"xG-%#lnG
6#*&,&\"#>\"\"\"-%%sqrtG6#\"$X%F+F+\"\"'!\"\"" }{TEXT -1 4 " et " }
{XPPEDIT 18 0 "y = ln((23-sqrt(445))/6);" "6#/%\"yG-%#lnG6#*&,&\"#B\"
\"\"-%%sqrtG6#\"$X%!\"\"F+\"\"'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2) " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve(4*arct
an(1/x)+arctan(1/y) = Pi/4); " }{TEXT -1 49 "Maple ne sait pas r\351so
udre l'\351quation directement" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%
#x|irG*&\"\"\"F'-%$cotG6#,&-%'arctanG6#*&F'F'%#y|irG!\"\"#F'\"\"%*&#\"
\"(\"#;F'%#PiGF'F'F1/F0F0" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 210 1 " " }{TEXT -1 26 "Cette \351quation entra\356ne \+
: " }{XPPEDIT 18 0 "tan(4*arctan(1/x)+arctan(1/y)) = 1;" "6#/-%$tanG6#
,&*&\"\"%\"\"\"-%'arctanG6#*&F*F*%\"xG!\"\"F*F*-F,6#*&F*F*%\"yGF0F*F*
" }{TEXT 210 1 " " }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "membregauche:=simplify(expa
nd(tan(4*arctan(1/x)+arctan(1/y))));  " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%-membregaucheG*&,,*&)%#x|irG\"\"$\"\"\"%#y|irGF+\"\"%*(F-F+F)F
+F,F+!\"\"*$)F)F-F+F+*&\"\"'F+)F)\"\"#F+F/F+F+F+,,*&F,F+F1F+F+*(F3F+F,
F+F4F+F/F,F+*&F-F+F(F+F/*&F-F+F)F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "solve(membregauche=1,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,,*$)%#x|irG\"\"%\"\"\"F)*&\"\"'F))F'\"\"#F)!\"\"F)F)
*&F(F))F'\"\"$F)F)*&F(F)F'F)F.F),,*$F0F)!\"%*&F(F)F'F)F)F%F)*&F+F)F,F)
F.F)F)F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
66 "Voici le script d'une proc\351dure destin\351e \340 trouver les va
leurs de " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 41 " entier naturel
 non nul pour lesquelles  " }{XPPEDIT 18 0 "y = (x^4-6*x^2+1+4*x^3-4*x
)/(-4*x^3+4*x+x^4-6*x^2+1);" "6#/%\"yG*&,,*$%\"xG\"\"%\"\"\"*&\"\"'F**
$F(\"\"#F*!\"\"F*F**&F)F**$F(\"\"$F*F**&F)F*F(F*F/F*,,*&F)F**$F(F2F*F/
*&F)F*F(F*F**$F(F)F**&F,F**$F(F.F*F/F*F*F/" }{TEXT -1 32 "  est un ent
ier relatif non nul." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 214 "cherche_solutions:=proc(n)\nlocal x;\nfo
r x from 1 to n do \nif \ntype((x^4-6*x^2+1+4*x^3-4*x)/(-4*x^3+4*x+x^4
-6*x^2+1),integer)=true then print([x,(x^4-6*x^2+1+4*x^3-4*x)/(-4*x^3+
4*x+x^4-6*x^2+1)])\nfi;\nod ;\nend proc;\n" }{TEXT -1 0 "" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%2cherche_solutionsGf*6#%\"nG6#%\"xG6\"F*?(8$
\"\"\"F-9$%%trueG@$/-%%typeG6$*&,,*$)F,\"\"%F-F-*&\"\"'F-)F,\"\"#F-!\"
\"F-F-*&F9F-)F,\"\"$F-F-*&F9F-F,F-F>F-,,*$F@F-!\"%*&F9F-F,F-F-F7F-*&F;
F-F<F-F>F-F-F>%(integerGF/-%&printG6#7$F,F5F*F*F*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "cherche_solutions(1000);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"&
!$R#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "
On peut v\351rifier que le couple " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG
\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }
{TEXT -1 74 " n'est pas solution de l'\351quation initiale. Il n'exist
e qu'une solution : " }{XPPEDIT 18 0 "x = 5;" "6#/%\"xG\"\"&" }{TEXT 
-1 3 " , " }{XPPEDIT 18 0 "y = -239;" "6#/%\"yG,$\"$R#!\"\"" }{TEXT 
-1 6 " pour " }{XPPEDIT 18 0 "x <= 1000;" "6#1%\"xG\"%+5" }{TEXT -1 2 
". " }}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }