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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 256 36 "Equations differentielles ordinaires" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 30 "Solutions des exercices
 1 et 2" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 
-1 24 "Version du 12 avril 2002" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 259 "" 0 "" {TEXT -1 11 "Exercice 1." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "La procedure suivante, q
uoique un peu rustique, permet de visualiser l'allure des solutions. E
lle utilise une premiere procedure qui echange l'ordre des solutions s
i necessaire." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "ex:=proc(s
ols)\nlocal newsol;\nnewsol:=[op(sols)];\nif (lhs(sols[1])=y(t)) then
\nnewsol[1]:=sols[2];\nnewsol[2]:=sols[1];\nfi;newsol;\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%#exGf*6#%%solsG6#%'newsolG6\"F*C%>8$7#-%#o
pG6#9$@$/-%$lhsG6#&F26#\"\"\"-%\"yG6#%\"tGC$>&F-F9&F26#\"\"#>&F-FCF8F-
F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Puis on dessine 4 solu
tions, en prenant comme conditions initiales t=0, et un point dans cha
cun des cadrans." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 632 "calin := proc(
eq1,eq2,a,b)\nlocal e1,e2,s1,s2,s3,s4,p1,p2,p3,p4;\ne1:=diff(x(t),t)=e
q1;\ne2:=diff(y(t),t)=eq2;\ns1:=ex(dsolve(\{e1,e2,x(0)=1,y(0)=1\},\{x(
t),y(t)\}));\ns2:=ex(dsolve(\{e1,e2,x(0)=-1,y(0)=1\},\{x(t),y(t)\}));
\ns3:=ex(dsolve(\{e1,e2,x(0)=-1,y(0)=-1\},\{x(t),y(t)\}));\ns4:=ex(dso
lve(\{e1,e2,x(0)=1,y(0)=-1\},\{x(t),y(t)\}));\np1:=plot([rhs(s1[1]),rh
s(s1[2]),t=a..b],color=red,scaling=constrained):p2:=plot([rhs(s2[1]),r
hs(s2[2]),t=a..b],color=blue,scaling=constrained):p3:=plot([rhs(s3[1])
,rhs(s3[2]),t=a..b],scaling=constrained,color=black):p4:=plot([rhs(s4[
1]),rhs(s4[2]),t=a..b],scaling=constrained,color=green);\ndisplay(p1,p
2,p3,p4);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&calinGf*6&%$eq1G
%$eq2G%\"aG%\"bG6,%#e1G%#e2G%#s1G%#s2G%#s3G%#s4G%#p1G%#p2G%#p3G%#p4G6
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%(scalingG%,constrainedG>8+-Fap6%7%-Fep6#&FgnFhp-Fep6#&FgnF\\qF^q/Fcq%
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FfrFcsF6F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with(plots)
:with(plottools):#si necessaire" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warnin
g, the name arrow has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 
"Warning, the name arrow has been redefined\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "calin(-x(t)-2*y(t),4*x(t)-5*y(t),0,1);" }}{PARA 
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2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 260 "" 0 "" {TEXT -1 27 "Voyons les valurs propres :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(linalg):#si necessaire
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eigenvalues([[-1,-2],[4,-5]]);
" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and
 trace have been redefined and unprotected\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$^$!\"$\"\"#^$F$!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 89 "et on constate avec satisfaction qu'elles ont toutes deux des p
arties reelles negatives. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "calin(-3*x(t)+4*y(t),-2*x(t)+3*y(t),-1,1);" }}{PARA 13 "" 1 "" 
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1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv
e 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Ici, on n'y voit rien. Ca
lculons les valeurs propres et vecteurs propres (tapez ?eigenvalues ):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eigenvalues([[-3,4],[-2
,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eigenvectors([[-3,4],[-2,3]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"\"F$<#-%'vectorG6#7$F$F$7%!\"\"F
$<#-F'6#7$\"\"#F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 "Les vecteur
s propres sont donc v1=[2,1], correspondant \210 la valeur propre -1, \+
avec multiplicit\216e1, et v2=[1,1], correspondant a la valeur propre \+
1, avec multiplicite 1. On va changer notre procedure en prenant comme
 valeurs initiales, pour t=0, v1+v2, puis v1-v2, puis -v1-v2, puis -v1
+v2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "recalin := proc(e
q1,eq2,a,b,v1,v2)\nlocal e1,e2,s1,s2,s3,s4,p1,p2,p3,p4;\ne1:=diff(x(t)
,t)=eq1;\ne2:=diff(y(t),t)=eq2;\ns1:=ex(dsolve(\{e1,e2,x(0)=(v1+v2)[1]
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)[1],y(0)=(v1-v2)[2]\},\{x(t),y(t)\}));\ns3:=ex(dsolve(\{e1,e2,x(0)=(-
v1+v2)[1],y(0)=(-v1+v2)[2]\},\{x(t),y(t)\}));\ns4:=ex(dsolve(\{e1,e2,x
(0)=(-v1-v2)[1],y(0)=(-v1-v2)[2]\},\{x(t),y(t)\}));\np1:=plot([rhs(s1[
1]),rhs(s1[2]),t=a..b],color=red,scaling=constrained):p2:=plot([rhs(s2
[1]),rhs(s2[2]),t=a..b],color=blue,scaling=constrained):p3:=plot([rhs(
s3[1]),rhs(s3[2]),t=a..b],scaling=constrained,color=black):p4:=plot([r
hs(s4[1]),rhs(s4[2]),t=a..b],scaling=constrained,color=green);\ndispla
y(p1,p2,p3,p4);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "re
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)=.sIMebFho-Fh\\l6&Fj\\lF^]lF[]lF^]l-%(SCALINGG6#%,CONSTRAINEDG-%+AXES
LABELSG6%Q!6\"F]do-%%FONTG6#%(DEFAULTG-%%VIEWG6$FbdoFbdo" 1 2 0 1 10 
0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 
"" 0 "" {TEXT -1 11 "Exercice 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 50 "D'abord une procedure utile; les parametr
es sont :" }}{PARA 0 "" 0 "" {TEXT -1 78 "eq1,eq2 : partie de droite d
e la premiere, respectivement la deuxieme equation" }}{PARA 0 "" 0 "" 
{TEXT -1 63 "x0,y0 : coordonnee x, respectivement y de la condition in
itiale" }}{PARA 0 "" 0 "" {TEXT -1 49 "a,b : temps de depart et d'arri
vee respectivement" }}{PARA 0 "" 0 "" {TEXT -1 14 "coul : couleur" }}
{PARA 0 "" 0 "" {TEXT -1 176 "npoints : nombre de points de la traject
oire qui seront calcules. Plus (b-a) est grand, plus il faudra calcule
r de points pour avoir une bonne approximation de la trajectoire. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
247 "palin:=proc(eq1,eq2,x0,y0,a,b,coul,npoints)\nlocal e1,e2,ini;\ne1
:=diff(x(t),t)=eq1;\ne2:=diff(y(t),t)=eq2;\nini:=x(0)=x0,y(0)=y0;\node
plot(dsolve(\{e1,e2,ini\},\{x(t),y(t)\},numeric),[x(t),y(t)],a..b,scal
ing=constrained,color=coul,numpoints=npoints);\nend:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 38 "Cherchons les points critiques de a) :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sols:=solve(\{-x+y^2,-y+x^2
\},\{x,y\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG6%<$/%\"xG\"\"
!/%\"yGF)<$/F(\"\"\"/F+F.<$/F+-%'RootOfG6$,(*$)%#_ZG\"\"#F.F.F8F.F.F./
%&labelG%%_L11G/F(,&!\"\"F.F2F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
23 "Pour y voir plus clair:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "allvalues(sols[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yG,&#
!\"\"\"\"#\"\"\"*&^##F*F)F*-%%sqrtG6#\"\"$F*F*/%\"xG,&F'F**&^#F'F*F.F*
F*<$/F3F&/F%F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Les racines de
 z^2+z+1 ne sont pas reelles; il y a donc 2 points critiques reels (0,
0) et (1,1). En (0,0) la partie lineaire est (-x,-y), donc pres de (0,
0) les trajectoires reviennent vers ce point. " }}{PARA 0 "" 0 "" 
{TEXT -1 127 "On trace quelques trajectoires pres de (1,1); apres tato
nnement, on arrive a choisir des valeurs de x0,y0, a et b convenables \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "p1:=palin(-x(t)+y(t)^
2,-y(t)+x(t)^2,0.9,1.2,0,1,red,10):p2:=palin(-x(t)+y(t)^2,-y(t)+x(t)^2
,0.7,1.2,0,1,blue,10):p3:=palin(-x(t)+y(t)^2,-y(t)+x(t)^2,0.9,0.8,0,1,
green,10):p4:=palin(-x(t)+y(t)^2,-y(t)+x(t)^2,1.2,0.9,0,1,brown,10):di
splay(p1,p2,p3,p4,pointplot([[0.9,1.2],[0.7,1.2],[0.9,0.8],[1.2,0.9]],
color=black,symbol=circle));" }}{PARA 13 "" 1 "" {GLPLOT2D 398 298 
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FJFH7$FOFM7$FTFR7$FYFW7$FhnFfn7$F]oF[o7$FboF`o7$FgoFeo7$F\\pFjo7$FapF_
p7$FfpFdp7$F[qFip7$F`qF^q7$FeqFcq7$FjqFhq7$F_rF]r7$FdrFbr7$FirFgr7$F^s
F\\s7$FcsFas7$FhsFfs7$F]tF[t7$FbtF`t7$FgtFet7$F\\uFjt7$FauF_u7$FfuFdu7
$F[vFiu7$F`vF^v7$FevFcv7$FjvFhv-F]w6&F_w$\")#)eqkFbw$\"))eqk\"FbwFe`m-
%'POINTSG6(7$$\"\"*!\"\"$\"#7F]am7$$\"\"(F]amF^am7$F[am$\"\")F]am7$F^a
mF[am-F]w6&F_wFdwFdwFdw-%'SYMBOLG6#%'CIRCLEG-%(SCALINGG6#%,CONSTRAINED
G-%+AXESLABELSG6%Q\"x6\"Q\"yFebm-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 
9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "ce qui \+
suggere que (1,1) est un point critique de type \"selle\" : 2 valeurs \+
propres reelles, de signe oppose. Voici encore, pour aider a comprendr
e, des trajectoires partant de divers points de l'axe OX et de l'axe O
Y :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "p1:=palin(-x(t)+y(t
)^2,-y(t)+x(t)^2,0,1.2,0,10,red,50):p2:=palin(-x(t)+y(t)^2,-y(t)+x(t)^
2,0,1.7,0,3,blue,20):p3:=palin(-x(t)+y(t)^2,-y(t)+x(t)^2,0,1.8,0,3,gre
en,20):p4:=palin(-x(t)+y(t)^2,-y(t)+x(t)^2,1.7,0,0,2,brown,20):p5:=pal
in(-x(t)+y(t)^2,-y(t)+x(t)^2,1.8,0,0,1.5,black,20):display(p1,p2,p3,p4
,p5,pointplot([[0,1.2],[0,1.7],[0,1.8],[1.7,0],[1.8,0]],color=black,sy
mbol=circle));" }}{PARA 13 "" 1 "" {GLPLOT2D 398 298 298 {PLOTDATA 2 "
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