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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 25 "Equations diff\351renti
elles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 1" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "dsolve(D(y)(t)+2*y(t)=t*cos(t),y(t));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,,*&F'\"\"\"-%$cosGF&F*#\"\"#\"\"&*&
#\"\"$\"#DF*F+F*!\"\"*(#F*F/F*-%$sinGF&F*F'F*F**&#\"\"%F3F*F7F*F4*&-%$
expG6#,$F'!\"#F*%$_C1GF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "dsolve(D(y)(t)+y(t)=exp(t)*cos(t),y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&-%$expGF&\"\"\"-%$cosGF&F,#\"\"#\"\"&
*(#F,F1F,-%$sinGF&F,F*F,F,*&-F+6#,$F'!\"\"F,%$_C1GF,F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dsolve(D(y)(t)+2*t*y(t)=t^2*exp(t),
y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&F'\"\"\"-%
$expGF&F*#F*\"\"#*&#F*\"\"%F*F+F*!\"\"**^##F*\"\")F*-%%sqrtG6#%#PiGF*-
%$erfG6#,&*&^#F*F*F'F*F*^#F-F*F*-F,6#,&*$)F'F.F*F2#F*F1F2F*F**&-F,6#,$
FEF2F*%$_C1GF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "?erf" }}
}{EXCHG {PARA 15 "" 0 "" {TEXT -1 51 "The error function is defined fo
r all complex x by " }}{PARA 0 "" 0 "" {TEXT 23 50 "      erf(x) = 2/s
qrt(Pi) * int(exp(-t^2), t=0..x)" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 61 "dsolve(\{D(y)(t)-exp(t)*y(t)=2*t*exp(exp(t)),y
(0)=1515\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&,
&*$)F'\"\"#\"\"\"F-*&\"%::F--%$expG6#F-!\"\"F-F--F16#-F1F&F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dsolve(t^3*D(y)(t)=2*y(t),y(
t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&%$_C1G\"\"\"-%
$expG6#,$*&F*F**$)F'\"\"#F*!\"\"F3F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 19 "Il faut recoller..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "dsolve(3*t*D(y)(t)-4*y(t)=t,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#%\"tG,&F'!\"\"*&)F'#\"\"%\"\"$\"\"\"%$_C1GF/F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Idem..." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "dsolve(t^2*D(y)(t)-(2*t-1)*y(t)=t^2,y(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*$)F'\"\"#\"\"\"F,*(F*
F,-%$expG6#*&F,F,F'!\"\"F,%$_C1GF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 19 "Une fois de plus..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "E
xercice 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(D(D(y))(
t)+y(t)=4*t^2*exp(t),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG
6#%\"tG,(*&-%$sinGF&\"\"\"%$_C2GF,F,*&-%$cosGF&F,%$_C1GF,F,*(\"\"#F,),
&F'F,F,!\"\"F3F,-%$expGF&F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "dsolve(D(D(y))(t)+y(t)=t*cos(t),y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&-%$sinGF&\"\"\"%$_C2GF,F,*&-%$cosGF&F
,%$_C1GF,F,*(#F,\"\"%F,F'F,,&F/F,*&F'F,F*F,F,F,F," }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 56 "dsolve(D(D(y))(t)+2*D(y)(t)+2*y(t)=exp(-t)*c
os(t),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*(-%$e
xpG6#,$F'!\"\"\"\"\"-%$sinGF&F/%$_C2GF/F/*(F*F/-%$cosGF&F/%$_C1GF/F/*(
#F/\"\"#F/F*F/,&F4F/*&F'F/F0F/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "dsolve(D(D(y))(t)-4*D(y)(t)+3*y(t)=t*exp(t)+cos(t),y(
t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,,*&-%$expGF&\"
\"\"%$_C2GF,F,*&-F+6#,$F'\"\"$F,%$_C1GF,F,*(#F,\"#SF,,(*$)F'\"\"#F,!#5
*&\"#5F,F'F,!\"\"\"\"&F>F,F*F,F,*&#F,F=F,-%$cosGF&F,F,*&#F,F?F,-%$sinG
F&F,F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dsolve(D(D(y))(t)
-2*D(y)(t)+5*y(t)=2*exp(t)*cos(t)^2,y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,(*(-%$expGF&\"\"\"-%$sinG6#,$F'\"\"#F,%$
_C2GF,F,*(F*F,-%$cosGF/F,%$_C1GF,F,*(#F,\"#;F,F*F,,(F4F,*(\"\"%F,F-F,F
'F,F,F<F,F,F," }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 3" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dsolve(t^2*D(D(y))(t)+t*D(y)
(t)-9*y(t)=6*t^3,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%
\"tG,(*&)F'\"\"$\"\"\"%$_C2GF,F,*&F'!\"$%$_C1GF,F,*(#F,\"\"'F,F*F,,&-%
#lnGF&F3F,!\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dsol
ve(t*D(D(y))(t)-D(y)(t)-t^3*y(t)=0,y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&%$_C1G\"\"\"-%%sinhG6#,$*$)F'\"\"#F+#
F+F2F+F+*&%$_C2GF+-%%coshGF.F+F+" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 10 "Exercice 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(
D(y)(t)=exp(y(t)),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%
\"tG-%#lnG6#,$*&\"\"\"F-,&F'F-%$_C1GF-!\"\"F0" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "dsolve(D(y)(t)=1+y(t)^2,y(t));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"yG6#%\"tG-%$tanG6#,&F'\"\"\"%$_C1GF," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve(D(y)(t)=1+y(t)^4+abs(
y(t))^(3/2),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"tG\"\"\"-%
$IntG6$*&F&F&,(F&F&*$)%#_aG\"\"%F&F&*$)-%$absG6#F.#\"\"$\"\"#F&F&!\"\"
/F.;%!G-%\"yG6#F%F8%$_C1GF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
88 "Maple ne trouve pas d'expression simple pour y(t), mais une relati
on  v\351rifi\351e par y(t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "?dsolve" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 13 "Invoking the " }
{TEXT 35 6 "dsolve" }{TEXT -1 19 " function with the " }{TEXT 35 7 "nu
meric" }{TEXT -1 4 " or " }{TEXT 35 12 "type=numeric" }{TEXT -1 15 " o
ption causes " }{TEXT 35 6 "dsolve" }{TEXT -1 82 " to find a numerical
 solution for the ODE or ODE system. If the optional equation " }
{TEXT 35 7 "method=" }{TEXT 40 13 "numericmethod" }{TEXT -1 20 " is pr
ovided (where " }{TEXT 40 13 "numericmethod" }{TEXT -1 11 " is one of \+
" }{HYPERLNK 17 "rkf45" 2 "dsolve[rkf45]" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "rosenbrock" 2 "dsolve[rosenbrock]" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "bvp" 2 "dsolve[numeric,bvp]" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "dverk78" 2 "dsolve[dverk78]" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "lsode" 2 "dsolve[lsode]" "" }{TEXT -1 2 ", " }{HYPERLNK 
17 "gear" 2 "dsolve[gear]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "taylorse
ries" 2 "dsolve[taylorseries]" "" }{TEXT -1 5 ", or " }{HYPERLNK 17 "c
lassical" 2 "dsolve[classical]" "" }{TEXT -1 3 "), " }{TEXT 35 6 "dsol
ve" }{TEXT -1 51 " uses that method to obtain the numerical solution.
" }}{PARA 14 "" 0 "" {TEXT -1 134 "Both initial and boundary value pro
blems can be numerically solved. For a system input, the type of probl
em is automatically detected." }}{PARA 14 "" 0 "" {TEXT -1 126 "By def
ault, a procedure is returned that can be used to obtain solution valu
es if given the value of the independent variable." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f:=dsolve(
\{D(y)(t) = 1+y(t)^4+abs(y(t))^(3/2), y(0)=0\}, y(t),type=numeric);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%(rkf45_xG6.%\"iG%/comp_sol
n_dataG%(odeprocG%&icvecG%(LR_caseG%\"YG%$valG%)outpointG%#ptG%*stop_p
rocG%+stop_arrayG%&cplexG6#%aoCopyright~(c)~2000~by~the~University~of~
Waterloo.~All~rights~reserved.GE\\s\"Q(complex6\"%&falseGC2@$-%'member
G6$9$7$Q4left_comp_soln_dataF9Q5right_comp_soln_dataF9O\"\"!@$2%'Digit
sG-%&truncG6#-%'evalhfG6#FH>FHFI>8+-%&evalfG6#F@@$4-%%typeG6$FQ.%(nume
ricG@%-F>6$FQ7%Q&startF9Q%leftF9Q&rightF9O$FEFEO-.9!FT>8&f*6&%\"NG%\"t
G%\"YG%#YPGF96#%.[Y[1]~=~y(t)]GF9>&9'6#\"\"\",(FapFap*$)&9&F`p\"\"%Fap
Fap*$)-%$absG6#Fep#\"\"$\"\"#FapFapF9F9F9>8'=F96#;FapFapE\\[l\"FapF^o>
8-FE>8.FE>8)-%$mapG6$%$rhsG-%&parseG6#-%#opG6$F^q-%%evalG6#Fdo>8/-Fbo6
#F8@'/FQF^oO7$/FhoFQ-%$seqG6$/&F[r6#8$-FS6#&FaqFgs/Fhs;Fap-%%nopsG6#F[
r1F^oFQ>8(FC>FctFB>8%-Fbo6#Fct@%4-FX6$7#Fft7#.%%listGC%Z%>Fft-%;dsolve
/numeric/SC/IVPsolveG6)Fdo;F^oFQFaqFgqFiqFjr/%'scprocGF:F9C&@$/Fjr%%tr
ueGYF9>FjrF]v>FftFdu>F[sF]v>8*&&Fft6#F^q6#&Fft6#F_q>FgtFftC$>FcvFdv@$5
3/FctFC2FcvFQ3/FctFB2FQFcvC&@$5/&Fft6#\"\")Fap2\"#5FiwC$>FcvFdv@%F`wY6
$QQcannot~evaluate~the~solution~further~right~of~%1F9-&FSFjw6#FcvY6$QP
cannot~evaluate~the~solution~further~left~of~%1F9FdxZ%>Fft-%<dsolve/nu
meric/SC/IVPextendG6*FftFdoFguFaqFgqFiqFjrFhuF9C&@$F\\vYF9>FjrF]v>FftF
\\y>F[sF]v>FgtFft>FcvFdv@&F`wC$@$33/%:_Env_smart_dsolve_numericGF]v2F^
oFcv2-Fbo6#F\\oFcv>F`zFcv@$Faw@)/&FgtFjwFapY6$Qcocannot~evaluate~the~s
olution~further~right~of~%1,~probably~a~singularityF9Fdx/FfzF_qY6$Qcqc
annot~evaluate~the~solution~further~right~of~%1,~maxfun~limit~exceeded
~(see~?dsolve,maxfun~for~details)F9Fdx2F]xFfzC$@$0%7_Env_dsolve_nowarn
stopGF]v-%(WARNINGG6#-_%,StringToolsG%.FormatMessageG6%Qhocannot~evalu
ate~the~solution~further~right~of~%1,~stop~condition~#%2~violatedF9Fdx
,&FfzFapF]x!\"\">FQFcvYFbxFcwC$@$33F\\z2FcvF^o2Fcv-Fbo6#F[o>Ff\\lFcv@$
Fdw@)FezY6$Qbocannot~evaluate~the~solution~further~left~of~%1,~probabl
y~a~singularityF9FdxFjzY6$Qbqcannot~evaluate~the~solution~further~left
~of~%1,~maxfun~limit~exceeded~(see~?dsolve,maxfun~for~details)F9FdxF^[
lC$@$Fa[l-Fd[l6#-Fg[l6%Qgocannot~evaluate~the~solution~further~left~of
~%1,~stop~condition~#%2~violatedF9FdxF\\\\l>FQFcvYFhx>Fcv-%9dsolve/num
eric/SC/IVPvalG6$FftFQ7$Fas-Fcs6$/Ffs&FcvFgsF\\tF9F9F9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(0);f(0.5);f(1);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7$/%\"tG$\"\"!F'/-%\"yG6#F%F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$/%\"tG$\"\"&!\"\"/-%\"yG6#F%$\"3T$)R!*3\\BEf!#=" }}
{PARA 8 "" 1 "" {TEXT -1 94 "Error, (in f) cannot evaluate the solutio
n further right of .87965690, probably a singularity\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?odeplot" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 35 7 "odeplot" }{TEXT -1 108 " function plots
 a solution curve (either 2-D or 3-D) determined from the output of a \+
call to dsolve/numeric." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "
with(plots):p:= a-> dsolve(\{D(y)(t) = 1+y(t)^4+abs(y(t))^(3/2), y(0)=
a\}, y(t),type=numeric):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+
name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "display(odeplot(p(0),[t,y(t)],-1..1),odeplot(p(2),[t,
y(t)],-3..1));" }}{PARA 7 "" 1 "" {TEXT -1 70 "Warning, cannot compute
 solution further left of -.879657912568074684\n" }}{PARA 7 "" 1 "" 
{TEXT -1 70 "Warning, cannot compute solution further right of .879657
912568074684\n" }}{PARA 7 "" 1 "" {TEXT -1 70 "Warning, cannot compute
 solution further left of -1.72209654242706978\n" }}{PARA 7 "" 1 "" 
{TEXT -1 73 "Warning, cannot compute solution further right of .372319
357082540950e-1\n" }}{PARA 13 "" 1 "" {GLPLOT2D 446 160 160 {PLOTDATA 
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