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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 73 "Equations diff\351rentiel
les - Tangentes horizontales et points d'inflexion " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 45 "II- R\351solution d'une
 \351quation diff\351rentielle " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 123 "Eq :=diff(y(x),x)-2*x*y(x)=x ;     # On stocke l'\351quation \+
dans Eq\ndsolve(Eq,y(x)) ;                  # On r\351sout l'\351quati
on" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EqG/,&-%%diffG6$-%\"yG6#%\"xG
F-\"\"\"*(\"\"#F.F-F.F*F.!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "sol:=rhs(dsolve(Eq,y(x)));
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 le membre de gauche de l'\351galit\351 pr\351c\351dente" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$solG,&#\"\"\"\"\"#!\"\"*&-%$expG6#*$)%\"xGF(F
'F'%$_C1GF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "sol :=subs
( C1=lambda, sol) ;      # On remplace le param\350tre _C1 par lambda
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,&#\"\"\"\"\"#!\"\"*&-%$exp
G6#*$)%\"xGF(F'F'%$_C1GF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
67 "dsolve(\{Eq,y(0)=1\},y(x)) ;         # On r\351sout le syst\350me \+
de Cauchy" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&#\"\"\"\"
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{TEXT 260 23 "III \023Etude d'un exemple" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart;\nwith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 
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bip$\"3_$**)pT<T2uF`oFg\\_l-%+AXESLABELSG6%Q\"x6\"Q\"yFde^p-%%FONTG6#%
(DEFAULTG-%%VIEWG6$;F(FbipF]f^p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 18 "IV- G\351
n\351ralisation" }}{PARA 0 "" 0 "" {TEXT -1 142 "Nos commandes pr\351c
\351dentes sont ind\351pendantes de l'\351quation de d\351part. Il nou
s suffit de les recopier et de les incorporer dans les proc\351dures.
" }}{PARA 0 "" 0 "" {TEXT -1 4 "1)a)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 242 "graphe1:=proc(Eq,x0)\n         local Sol,L;\nSol:=rh
s(dsolve(\{Eq,y(x0)=a/2\},y(x)));\nL:=[seq(eval(Sol),a=-10..10)];     \+
      # Nous ajoutons la commande eval pour forcer Maple \340 \351valu
er la valeur de la variable Sol\nplot(L,x=-5..5,y=-5..5);\nend;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(graphe1Gf*6$%#EqG%#x0G6$%$SolG%\"LG
6\"F,C%>8$-%$rhsG6#-%'dsolveG6$<$9$/-%\"yG6#9%,$*&\"\"#!\"\"%\"aG\"\"
\"FB-F:6#%\"xG>8%7#-%$seqG6$-%%evalG6#F//FA;!#5\"#5-%%plotG6%FG/FE;!\"
&\"\"&/F:FWF,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "1)b) " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "graphe2:=proc(Eq)\n        \+
 local Eq1,TH;\nEq1:=subs(diff(y(x),x)=0,Eq); \nTH:=solve(Eq1,y(x));\n
plot(TH,x=-5..5,y=-5..5,color=black,thickness=2);\nend;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(graphe2Gf*6#%#EqG6$%$Eq1G%#THG6\"F+C%>8$-%%su
bsG6$/-%%diffG6$-%\"yG6#%\"xGF9\"\"!9$>8%-%&solveG6$F.F6-%%plotG6'F=/F
9;!\"&\"\"&/F7FE/%&colorG%&blackG/%*thicknessG\"\"#F+F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "1)c)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 304 "graphe3:=proc(Eq)\n         local Eqp,Eq2,yprime,Eq3
,TI;\nEqp:=diff(Eq,x);\nEq2:=subs(diff(y(x),x$2)=0,Eqp);\nyprime:=solv
e(Eq2,diff(y(x),x));\nEq3:=subs(diff(y(x),x)=yprime,Eq);  # On remplac
e y'(x) par la valeur trouv\351e pr\351c\351demment\nTI:=solve(Eq3,y(x
));\nplot(TI,x=-5..5,y=-5..5,color=blue,thickness=2);\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%(graphe3Gf*6#%#EqG6'%$EqpG%$Eq2G%'yprimeG%
$Eq3G%#TIG6\"F.C(>8$-%%diffG6$9$%\"xG>8%-%%subsG6$/-F36$-%\"yG6#F6-%\"
$G6$F6\"\"#\"\"!F1>8&-%&solveG6$F8-F36$F?F6>8'-F:6$/FLFHF5>8(-FJ6$FOF?
-%%plotG6'FT/F6;!\"&\"\"&/F@Fen/%&colorG%%blueG/%*thicknessGFEF.F.F." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "1)d)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 121 "graphe:=proc(Eq,x0)\n        local G1,G2,G3;\nG1:=
graphe1(Eq,x0);\nG2:=graphe2(Eq);\nG3:=graphe3(Eq);\ndisplay(G1,G2,G3)
;\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'grapheGf*6$%#EqG%#x0G6%%
#G1G%#G2G%#G3G6\"F-C&>8$-%(graphe1G6$9$9%>8%-%(graphe2G6#F4>8&-%(graph
e3GF:-%(displayG6%F0F7F<F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "
2)a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "Eq:=diff(y(x),x)+x
*y(x)=x^3;\ngraphe(Eq,0);\ngraphe(Eq,1.5);   # En se pla\347ant en 1.5
, on observe des comportements plus vari\351s pour les solutions de l'
\351quation." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EqG/,&-%%diffG6$-%
\"yG6#%\"xGF-\"\"\"*&F-F.F*F.F.*$)F-\"\"$F." }}{PARA 13 "" 1 "" 
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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur
ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15
" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "C
urve 22" "Curve 23" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "2)b)\nDan
s ce cas, les intervalles de r\351solution de l'\351quation sont ]-oo,
0[ et ]0,+oo[. On se place donc en 1. Les expressions trouv\351es fonc
tionnent \351galement sur ]-oo,0[." }}}{EXCHG {PARA 0 "> " 0 "" 
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^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" 
"Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" 
"Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curv
e 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21
" "Curve 22" "Curve 23" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 245 "3)b)
\nLes intervalles de r\351solution de cette \351quation sont ]-oo,-1[,
 ]-1,1[ et ]1,+oo[. Cette fois, les solutions trouv\351es par Maple ne
 sont d\351finies que sur leurs intervalles respectifs. Nous tra\347on
s les solutions pour chacun de ces intervalles." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 160 "Eq:=(x^2-1)*diff(y(x),x)+y(x)=x^3;\nG11:=grap
he1(Eq,2):\nG12:=graphe1(Eq,0):\nG13:=graphe1(Eq,-2):\nG2:=graphe2(Eq)
:\nG3:=graphe3(Eq):\ndisplay(\{G11,G12,G13,G2,G3\});\n" }}{PARA 11 "" 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 26 "V- Pour aller plus
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{PARA 8 "" 1 "" {TEXT -1 35 "Error, (in plot) invalid arguments\n" }}
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=6Fchm-%*THICKNESSG6#\"\"#-%'COLOURG6&%$RGBGFd[nFd[nFd[n-%+AXESLABELSG
6$Q\"x6\"Q\"yFaio-%%VIEWG6$;$!\"&Fd[nFb[nFfio" 1 2 0 1 10 2 2 6 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 37 "Faisons la m\352me chose pour graphe3 : " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Eqp:=diff(Eq,x);\nEq2:=subs
(diff(y(x),x$2)=0,Eqp);\nyprime:=solve(Eq2,diff(y(x),x));\nEq3:=subs(d
iff(y(x),x)=yprime,Eq);  # On remplace y'(x) par la valeur trouv\351e \+
pr\351c\351demment\nTI:=solve(Eq3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$EqpG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F*F2
-F(6$F*F-F2F2,$*&\"\"$F2)F-F1F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$Eq2G/,$*(\"\"#\"\"\"-%\"yG6#%\"xGF)-%%diffG6$F*F-F)F),$*&\"\"$F))F-F
(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'yprimeG,$*&#\"\"$\"\"#\"\"
\"*&%\"xGF)-%\"yG6#F,!\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E
q3G/,&*&#\"\"$\"\"#\"\"\"*&%\"xGF*-%\"yG6#F-!\"\"F+F+*$)F.F*F+F+*$)F-F
)F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#TIG6%,&*&\"\"'!\"\",&*&\"$i
\"\"\"\")%\"xG\"\"#F-F)*&F(F-,&*&\"#[F-)F/\"\"*F-F)*&\"$H(F-)F/\"\"%F-
F-#F-F0F-#F-\"\"$F-*(F0F-F/F=F*#F)F=F-,(*&\"#7F)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-,(*&FBF)F*F<F)FCF)*(^##F)F0F
-F=F;FFF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Le probl\350me es
t le m\352me que pour graphe2 dans cet exemple. Cependant, remarquons \+
que la r\351solution de yprime pourrait g\351n\351rer plusieurs soluti
ons. Nous en tenons compte dans la solution suivante : " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "graphe3:=proc(Eq)\n         local \+
Eqp,Eq2,yprime,Eq3,TI,yp;\nEqp:=diff(Eq,x);\nEq2:=subs(diff(y(x),x$2)=
0,Eqp);\nyprime:=\{solve(Eq2,diff(y(x),x))\};\nTI:=NULL;\n#  Nous ajou
tons les solutions une par une en g\351n\351rant une s\351quence TI\nf
or yp in yprime do \n    Eq3:=subs(diff(y(x),x)=yp,Eq);\n    TI:=TI,so
lve(Eq3,y(x));\nod;\nplot([TI],x=-5..5,y=-5..5,color=blue,thickness=2,
numpoints=300);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(graphe3Gf*
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%plotG6(7#FQ/F7;!\"&\"\"&/FAF_o/%&colorG%%blueG/%*thicknessGFF/%*numpo
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