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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Euler's Method - Example \+
3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 32 "So
lve the initial value problem " }}{PARA 259 "" 0 "" {TEXT -1 61 "     \+
                    dy/dx = x^2+y^2 ,      y(0) = 0     " }}{PARA 260 
"" 0 "" {TEXT -1 165 "using Euler's method of approximation. As we do \+
not have the true solution for this DE, we will decreasing the step si
ze h and observe how the approximation behaves." }}{PARA 0 "" 0 "" 
{TEXT -1 19 "                   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "h:=0.1;  # the step size" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "oldx:=0;  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "oldy:=0;  # the initial condition" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "lastx:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"L:=[];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=[op(L),[oldx,
oldy]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "slope:=(x,y) -> \+
x^2+ y^2;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "while old
x < lastx do oldy:=oldy+h*slope(oldx,oldy); oldx:=oldx+h; L:=[op(L),[o
ldx,oldy]]; od;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"LG777$\"\"!F'7$$\"\"\"!\"\"F'7$$\"\"#F+$F*!\"$7$$\"
\"$F+$\"&,+&!\"(7$$\"\"%F+$\"+5+E+9!#67$$\"\"&F+$\"+Q2A-IF<7$$\"\"'F+$
\"+nSB6bF<7$$\"\"(F+$\"+oxgT\"*F<7$$\"\")F+$\"+ww^79!#57$$\"\"*F+$\"+Q
(pC2#FQ7$$\"#5F+$\"+Y5UDHFQ7$$\"#6F+$\"+H>+6SFQ7$$\"#7F+$\"+%H$)=Q&FQ7
$$\"#8F+$\"+s*H:6(FQ7$$\"#9F+$\"+d&osI*FQ7$$\"#:F+$\"+/@N87!\"*7$$\"#;
F+$\"+PWd&e\"F_p7$$\"#<F+$\"+n!zH4#F_p7$$\"#=F+$\"+0_.?GF_p7$$\"#>F+$
\"+h]HRRF_p7$$\"#?F+$\"+>'*4_eF_p" }}}{EXCHG {PARA 12 "" 1 "" {TEXT 
-1 66 "Note that we deleted the intermediate solution for easy referen
ce." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "picture1:=plot(L,style=point,symbol=diamond );" }}
{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "h:=0.01;  # the step size" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "oldx:=0;  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "oldy:
=0;  # the initial condition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 6 "L:=[];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=[op(L),[o
ldx,oldy]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "slope:=(x,y)
 -> x^2+ y^2;" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "while oldx < lastx do oldy:=oldy+h*
slope(oldx,oldy); oldx:=oldx+h; L:=[op(L),[oldx,oldy]]; od;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"LG7ew7$\"\"!F'7$$\"\"\"!\"#F'7$$\"\"#F+$
F*!\"'7$$\"\"$F+$\"*,+++&!#97$$\"\"%F+$\"+E+++9F67$$\"\"&F+$\"+A-++IF6
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\"\"*F+$\"+.K+S?FP7$$\"#5F+$\"+lt+]GFP7$$\"#6F+$\"+)[:+&QFP7$$\"#7F+$
\"+6..g]FP7$$\"#8F+$\"+:f0+lFP7$$\"#9F+$\"+m\")4!>)FP7$$\"#:F+$\"+Cl,:
5!#77$$\"#;F+$\"+Fo-S7F^p7$$\"#<F+$\"+/A/'\\\"F^p7$$\"#=F+$\"+&ek]y\"F
^p7$$\"#>F+$\"+]k44@F^p7$$\"#?F+$\"+L49qCF^p7$$\"#@F+$\"+\\>?qGF^p7$$
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$\"#QF+$\"+$p0!f<Fdu7$$\"#RF+$\"+.^r.>Fdu7$$\"#SF+$\"+;v<c?Fdu7$$\"#TF
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;qrZFFdu7$$\"#XF+$\"+6?2UHFdu7$$\"#YF+$\"+!fPa9$Fdu7$$\"#ZF+$\"+op-eLF
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ugqFdu7$$\"#hF+$\"+bHtDuFdu7$$\"#iF+$\"+1rM.yFdu7$$\"#jF+$\"+Hj$Q>)Fdu
7$$\"#kF+$\"+C-X(f)Fdu7$$\"#lF+$\"+R=W9!*Fdu7$$\"#mF+$\"+by1X%*Fdu7$$
\"#nF+$\"+'y)e*))*Fdu7$$\"#oF+$\"+=p#[.\"!#57$$\"#pF+$\"+&yP@3\"Fi`l7$
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`l7$$\"#wF+$\"+z62c9Fi`l7$$\"#xF+$\"+A8&f^\"Fi`l7$$\"#yF+$\"+I%Rvd\"Fi
`l7$$\"#zF+$\"+h!o3k\"Fi`l7$$\"#!)F+$\"+40(fq\"Fi`l7$$\"#\")F+$\"+W3)G
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Fi`l7$$\"$D\"F+$\"+a,OjsFi`l7$$\"$E\"F+$\"+blOsuFi`l7$$\"$F\"F+$\"+<G'
po(Fi`l7$$\"$G\"F+$\"+9AM2zFi`l7$$\"$H\"F+$\"+v#3P8)Fi`l7$$\"$I\"F+$\"
+&[viO)Fi`l7$$\"$J\"F+$\"+^+F0')Fi`l7$$\"$K\"F+$\"+C2$4&))Fi`l7$$\"$L
\"F+$\"+*p4N5*Fi`l7$$\"$M\"F+$\"+)etKO*Fi`l7$$\"$N\"F+$\"+![/0j*Fi`l7$
$\"$O\"F+$\"+(4,b!**Fi`l7$$\"$P\"F+$\"+0!e)=5!\"*7$$\"$Q\"F+$\"+@x+[5F
cfm7$$\"$R\"F+$\"+B\\.y5Fcfm7$$\"$S\"F+$\"+;v(*36Fcfm7$$\"$T\"F+$\"+Fe
(39\"Fcfm7$$\"$U\"F+$\"+/Gxt6Fcfm7$$\"$V\"F+$\"+IUr27Fcfm7$$\"$W\"F+$
\"+n*[FC\"Fcfm7$$\"$X\"F+$\"+<#H*y7Fcfm7$$\"$Y\"F+$\"+;3J;8Fcfm7$$\"$Z
\"F+$\"+eN&\\N\"Fcfm7$$\"$[\"F+$\"+\\:#\\R\"Fcfm7$$\"$\\\"F+$\"+5OGO9F
cfm7$$\"$]\"F+$\"+;P6z9Fcfm7$$\"$^\"F+$\"+!\\\"\\B:Fcfm7$$\"$_\"F+$\"+
`F]p:Fcfm7$$\"$`\"F+$\"+U,C<;Fcfm7$$\"$a\"F+$\"+*z.om\"Fcfm7$$\"$b\"F+
$\"+[@I=<Fcfm7$$\"$c\"F+$\"+rF&=x\"Fcfm7$$\"$d\"F+$\"+$R$eF=Fcfm7$$\"$
e\"F+$\"+/Ij&)=Fcfm7$$\"$f\"F+$\"+AJ:Y>Fcfm7$$\"$g\"F+$\"+U#4$4?Fcfm7$
$\"$h\"F+$\"+yCGv?Fcfm7$$\"$i\"F+$\"+_9FW@Fcfm7$$\"$j\"F+$\"+`W\\;AFcf
m7$$\"$k\"F+$\"+I>>#H#Fcfm7$$\"$l\"F+$\"+o$H;P#Fcfm7$$\"$m\"F+$\"+F15b
CFcfm7$$\"$n\"F+$\"+==$Ha#Fcfm7$$\"$o\"F+$\"+Se[NEFcfm7$$\"$p\"F+$\"+'
pnJt#Fcfm7$$\"$q\"F+$\"+`2VOGFcfm7$$\"$r\"F+$\"+ZTyXHFcfm7$$\"$s\"F+$
\"+*e,=1$Fcfm7$$\"$t\"F+$\"+z=8&=$Fcfm7$$\"$u\"F+$\"+I:^;LFcfm7$$\"$v
\"F+$\"+<+ycMFcfm7$$\"$w\"F+$\"+)H)*og$Fcfm7$$\"$x\"F+$\"+^9(zw$Fcfm7$
$\"$y\"F+$\"+SlFTRFcfm7$$\"$z\"F+$\"+[rHGTFcfm7$$\"$!=F+$\"+@lwIVFcfm7
$$\"$\"=F+$\"+3>s]XFcfm7$$\"$#=F+$\"+2Od!z%Fcfm7$$\"$$=F+$\"+iN>`]Fcfm
7$$\"$%=F+$\"+9-.U`Fcfm7$$\"$&=F+$\"+#3f7m&Fcfm7$$\"$'=F+$\"+EE)f,'Fcf
m7$$\"$(=F+$\"+'4*\\7kFcfm7$$\"$)=F+$\"+V&p'eoFcfm7$$\"$*=F+$\"+AqUktF
cfm7$$\"$!>F+$\"+we\\UzFcfm7$$\"$\">F+$\"+$GG%4')Fcfm7$$\"$#>F+$\"+P=8
(Q*Fcfm7$$\"$$>F+$\"+Gy^I5!\")7$$\"$%>F+$\"+F%R/9\"F\\hn7$$\"$&>F+$\"+
OKEu7F\\hn7$$\"$'>F+$\"+;/WS9F\\hn7$$\"$(>F+$\"+v)o<l\"F\\hn7$$\"$)>F+
$\"+<Q[G>F\\hn7$$\"$*>F+$\"++#4VI#F\\hn7$$\"$+#F+$\"+*Q`#RGF\\hn" }}}
{EXCHG {PARA 12 "" 1 "" {TEXT -1 210 "Maple goes through 200 steps to \+
generate the above array. All the intermediate values have been delete
d. Please note that it will take a fair amount of time if you intend t
o generate the same array with h=0.01." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "picture2:=plot(L,style=point,symbol=circle );" }}
{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "display(\{picture1,picture2\});" }}{PARA 13 "" 1 "" {INLPLOT "6$
-%'CURVESG6&777$\"\"!F(7$$\"1+++++++5!#;F(7$$\"1+++++++?F,$F+!#=7$$\"1
+++++++IF,$\"1+++++5+]F17$$\"1+++++++SF,$\"1+++5+E+9!#<7$$\"1+++++++]F
,$\"1+++Q2A-IF<7$$\"1+++++++gF,$\"1+++nSB6bF<7$$\"1+++++++qF,$\"1+++ox
gT\"*F<7$$\"1+++++++!)F,$\"1+++ww^79F,7$$\"1+++++++!*F,$\"1+++Q(pC2#F,
7$$\"\"\"F($\"1+++Y5UDHF,7$$\"1+++++++6!#:$\"1+++H>+6SF,7$$\"1+++++++7
Fhn$\"1,++%H$)=Q&F,7$$\"1+++++++8Fhn$\"1+++s*H:6(F,7$$\"1+++++++9Fhn$
\"1+++d&osI*F,7$$\"1+++++++:Fhn$\"1+++/@N87Fhn7$$\"1+++++++;Fhn$\"1+++
PWd&e\"Fhn7$$\"1+++++++<Fhn$\"1+++n!zH4#Fhn7$$\"1+++++++=Fhn$\"1+++0_.
?GFhn7$$\"1+++++++>Fhn$\"1+++h]HRRFhn7$$\"\"#F($\"1+++>'*4_eFhn-%'COLO
URG6&%$RGBG$\"#5!\"\"F(F(-%'SYMBOLG6#%(DIAMONDG-%&STYLEG6#%&POINTG-F$6
&7ewF'7$$F+F<F(7$$F/F<$F+!#@7$$F4F<$\"1+++5+++]F_s7$$F9F<$\"1+++E+++9!
#?7$$F?F<$\"1+++A-++IFhs7$$FDF<$\"1+++A6++bFhs7$$\"1,++++++qF<$\"1+++Z
T++\"*Fhs7$$FNF<$\"1+++V7++9!#>7$$FSF<$\"1+++.K+S?Fjt7$F*$\"1+++lt+]GF
jt7$$FgnF,$\"1+++)[:+&QFjt7$$F]oF,$\"1*****4JI+1&Fjt7$$FboF,$\"1+++:f0
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o-S7F17$$FfpF,$\"1+++/A/'\\\"F17$$F[qF,$\"1+++&ek]y\"F17$$F`qF,$\"1+++
]k44@F17$F.$\"1+++L49qCF17$$\"1+++++++@F,$\"1+++\\>?qGF17$$\"1+++++++A
F,$\"1+++IVG6LF17$$\"1+++++++BF,$\"1+++wRR&z$F17$$\"1+++++++CF,$\"1+++
E!QXK%F17$$\"1+++++++DF,$\"1+++U]s+\\F17$$\"1+++++++EF,$\"1+++8_'f_&F1
7$$\"1+++++++FF,$\"1+++w0F-iF17$$\"1+++++++GF,$\"1+++e_lJpF17$$\"1++++
+++HF,$\"1+++Od8;xF17$F3$\"1+++C6td&)F17$$\"1+++++++JF,$\"1+++sMYe%*F1
7$$\"1+++++++KF,$\"1+++5e.U5F<7$$\"1+++++++LF,$\"1+++%RWX9\"F<7$$\"1++
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$\"1+++W+-#\\\"F<7$$\"1+++++++PF,$\"1+++cE%=i\"F<7$$\"1+++++++QF,$\"1+
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++\\0;iDF<7$$\"1+++++++WF,$\"1+++;qrZFF<7$$\"1+++++++XF,$\"1+++6?2UHF<
7$$\"1+++++++YF,$\"1+++!fPa9$F<7$$\"1+++++++ZF,$\"1+++op-eLF<7$$\"1+++
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\"1+++++++`F,$\"1+++=5PR[F<7$$\"1+++++++aF,$\"1+++pHhA^F<7$$\"1+++++++
bF,$\"1+++&3PoT&F<7$$\"1,++++++cF,$\"1+++(HrAs&F<7$$\"1+++++++dF,$\"1+
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$FC$\"1******QvugqF<7$$\"1+++++++hF,$\"1+++bHtDuF<7$$\"1+++++++iF,$\"1
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F<7$$\"1+++++++lF,$\"1******Q=W9!*F<7$$\"1+++++++mF,$\"1+++by1X%*F<7$$
\"1+++++++nF,$\"1+++'y)e*))*F<7$$\"1+++++++oF,$\"1+++=p#[.\"F,7$$\"1**
************oF,$\"1+++&yP@3\"F,7$FH$\"1+++2)=48\"F,7$$\"1+++++++rF,$\"
1+++%y(>\"=\"F,7$$\"1+++++++sF,$\"1+++7I+L7F,7$$\"1+++++++tF,$\"1+++3L
O'G\"F,7$$\"1+++++++uF,$\"1+++R!38M\"F,7$$\"1+++++++vF,$\"1+++Yr'yR\"F
,7$$\"1+++++++wF,$\"1+++z62c9F,7$$\"1+++++++xF,$\"1+++A8&f^\"F,7$$\"1+
++++++yF,$\"1+++I%Rvd\"F,7$$\"1+++++++zF,$\"1+++h!o3k\"F,7$FM$\"1+++40
(fq\"F,7$$\"1,++++++\")F,$\"1+++W3)Gx\"F,7$$\"1+++++++#)F,$\"1+++]RjT=
F,7$$\"1+++++++$)F,$\"1+++mbE7>F,7$$\"1+++++++%)F,$\"1+++EB\"[)>F,7$$
\"1+++++++&)F,$\"1+++1=Jf?F,7$$\"1+++++++')F,$\"1+++rD!e8#F,7$$\"1++++
+++()F,$\"1+++CUK9AF,7$$\"1+++++++))F,$\"1+++cu\"\\H#F,7$$\"1+++++++*)
F,$\"1+++-TixBF,7$FR$\"1+++)>([iCF,7$$\"1+++++++\"*F,$\"1+++T5b\\DF,7$
$\"1+++++++#*F,$\"1+++_7')QEF,7$$\"1+++++++$*F,$\"1+++T[YIFF,7$$\"1***
**********R*F,$\"1+++z-TCGF,7$$\"1+++++++&*F,$\"1+++svu?HF,7$$\"1+++++
++'*F,$\"1+++Q$G&>IF,7$$\"1+++++++(*F,$\"1+++*)e!37$F,7$$\"1+++++++)*F
,$\"1+++=`jCKF,7$$\"1+++++++**F,$\"1+++\"ft5L$F,7$FW$\"1+++U'z,W$F,7$$
\"1++++++55Fhn$\"1+++yW,_NF,7$$\"1++++++?5Fhn$\"1+++&GTmm$F,7$$\"1++++
++I5Fhn$\"1+++Vb7%y$F,7$$\"1++++++S5Fhn$\"1+++\\^`/RF,7$$\"1++++++]5Fh
n$\"1+++W0%z-%F,7$$\"1++++++g5Fhn$\"1+++\\[TaTF,7$$\"1++++++q5Fhn$\"1+
++7S.%G%F,7$$\"1++++++!3\"Fhn$\"1+++fp(oT%F,7$$\"1++++++!4\"Fhn$\"1+++
hd-`XF,7$Ffn$\"1+++0ec#p%F,7$$\"1++++++56Fhn$\"1+++zfeN[F,7$$\"1++++++
?6Fhn$\"1+++r)y@)\\F,7$$\"1++++++I6Fhn$\"1+++x4WK^F,7$$\"1++++++S6Fhn$
\"1+++FHZ'G&F,7$$\"1++++++]6Fhn$\"1+++B(zVW&F,7$$\"1++++++g6Fhn$\"1+++
%*4F1cF,7$$\"1++++++q6Fhn$\"1+++p7EsdF,7$$\"1++++++!=\"Fhn$\"1+++p-ZUf
F,7$$\"1++++++!>\"Fhn$\"1+++AK-<hF,7$F\\o$\"1+++&>^gH'F,7$$\"1++++++57
Fhn$\"1+++c9pzkF,7$$\"1++++++?7Fhn$\"1+++dy3omF,7$$\"1++++++I7Fhn$\"1+
++`7RhoF,7$$\"1++++++S7Fhn$\"1+++V*f(fqF,7$$\"1++++++]7Fhn$\"1+++a,Ojs
F,7$$\"1++++++g7Fhn$\"1+++blOsuF,7$$\"1++++++q7Fhn$\"1+++<G'po(F,7$$\"
1++++++!G\"Fhn$\"1+++9AM2zF,7$$\"1++++++!H\"Fhn$\"1+++v#3P8)F,7$Fao$\"
1+++&[viO)F,7$$\"1++++++58Fhn$\"1+++^+F0')F,7$$\"1++++++?8Fhn$\"1+++C2
$4&))F,7$$\"1++++++I8Fhn$\"1+++*p4N5*F,7$$\"1++++++S8Fhn$\"1+++)etKO*F
,7$$\"1++++++]8Fhn$\"1******zW]I'*F,7$$\"1++++++g8Fhn$\"1+++(4,b!**F,7
$$\"1++++++q8Fhn$\"1+++0!e)=5Fhn7$$\"1++++++!Q\"Fhn$\"1+++@x+[5Fhn7$$
\"1++++++!R\"Fhn$\"1+++B\\.y5Fhn7$Ffo$\"1+++;v(*36Fhn7$$\"1++++++59Fhn
$\"1+++Fe(39\"Fhn7$$\"1++++++?9Fhn$\"1+++/Gxt6Fhn7$$\"1++++++I9Fhn$\"1
+++IUr27Fhn7$$\"1++++++S9Fhn$\"1+++n*[FC\"Fhn7$$\"1++++++]9Fhn$\"1+++<
#H*y7Fhn7$$\"1++++++g9Fhn$\"1+++;3J;8Fhn7$$\"1++++++q9Fhn$\"1+++eN&\\N
\"Fhn7$$\"1++++++![\"Fhn$\"1+++\\:#\\R\"Fhn7$$\"1++++++!\\\"Fhn$\"1+++
5OGO9Fhn7$F[p$\"1+++;P6z9Fhn7$$\"1++++++5:Fhn$\"1+++!\\\"\\B:Fhn7$$\"1
++++++?:Fhn$\"1+++`F]p:Fhn7$$\"1++++++I:Fhn$\"1+++U,C<;Fhn7$$\"1++++++
S:Fhn$\"1+++*z.om\"Fhn7$$\"1++++++]:Fhn$\"1+++[@I=<Fhn7$$\"1++++++g:Fh
n$\"1+++rF&=x\"Fhn7$$\"1++++++q:Fhn$\"1+++$R$eF=Fhn7$$\"1++++++!e\"Fhn
$\"1+++/Ij&)=Fhn7$$\"1++++++!f\"Fhn$\"1+++AJ:Y>Fhn7$F`p$\"1+++U#4$4?Fh
n7$$\"1++++++5;Fhn$\"1+++yCGv?Fhn7$$\"1++++++?;Fhn$\"1+++_9FW@Fhn7$$\"
1++++++I;Fhn$\"1+++`W\\;AFhn7$$\"1++++++S;Fhn$\"1+++I>>#H#Fhn7$$\"1+++
+++];Fhn$\"1+++o$H;P#Fhn7$$\"1++++++g;Fhn$\"1+++F15bCFhn7$$\"1++++++q;
Fhn$\"1+++==$Ha#Fhn7$$\"1++++++!o\"Fhn$\"1+++Se[NEFhn7$$\"1++++++!p\"F
hn$\"1+++'pnJt#Fhn7$Fep$\"1+++`2VOGFhn7$$\"1++++++5<Fhn$\"1+++ZTyXHFhn
7$$\"1++++++?<Fhn$\"1+++*e,=1$Fhn7$$\"1++++++I<Fhn$\"1+++z=8&=$Fhn7$$
\"1++++++S<Fhn$\"1+++I:^;LFhn7$$\"1++++++]<Fhn$\"1+++<+ycMFhn7$$\"1+++
+++g<Fhn$\"1+++)H)*og$Fhn7$$\"1++++++q<Fhn$\"1+++^9(zw$Fhn7$$\"1++++++
!y\"Fhn$\"1+++SlFTRFhn7$$\"1++++++!z\"Fhn$\"1+++[rHGTFhn7$Fjp$\"1+++@l
wIVFhn7$$\"1++++++5=Fhn$\"1+++3>s]XFhn7$$\"1++++++?=Fhn$\"1+++2Od!z%Fh
n7$$\"1++++++I=Fhn$\"1+++iN>`]Fhn7$$\"1++++++S=Fhn$\"1+++9-.U`Fhn7$$\"
1++++++]=Fhn$\"1+++#3f7m&Fhn7$$\"1++++++g=Fhn$\"1+++EE)f,'Fhn7$$\"1+++
+++q=Fhn$\"1+++'4*\\7kFhn7$$\"1++++++!)=Fhn$\"1+++V&p'eoFhn7$$\"1+++++
+!*=Fhn$\"1+++AqUktFhn7$F_q$\"1+++we\\UzFhn7$$\"1++++++5>Fhn$\"1+++$GG
%4')Fhn7$$\"1++++++?>Fhn$\"1+++P=8(Q*Fhn7$$\"1++++++I>Fhn$\"1+++Gy^I5!
#97$$\"1++++++S>Fhn$\"1+++F%R/9\"Fh[o7$$\"1++++++]>Fhn$\"1+++OKEu7Fh[o
7$$\"1++++++g>Fhn$\"1+++;/WS9Fh[o7$$\"1++++++q>Fhn$\"1+++v)o<l\"Fh[o7$
$\"1++++++!)>Fhn$\"1+++<Q[G>Fh[o7$$\"1++++++!*>Fhn$\"1++++#4VI#Fh[o7$F
dq$\"1+++*Q`#RGFh[oFhq-F`r6#%'CIRCLEGFcr" 2 304 304 304 2 0 1 0 2 9 0 
4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 
0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 7 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 126 "The graph shows that if the step size h is not smal
l enough, the approximation becomes unreliable very quickly as x incre
ases." }}{PARA 0 "" 0 "" {TEXT -1 77 "You can compare the solution wit
h the series approxiamtion obtained from SWP." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "26 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }