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kground = \"[255,255,255]\"), Typesetting:-mi(\"t\")), Typesetting:-mi
(\"\")), linethickness = \"1\", denomalign = \"center\", numalign = \"
center\", bevelled = \"false\", foreground = \"[0,0,0]\", background =
 \"[255,255,255]\"), Typesetting:-mo(\"&InvisibleTimes;\", form = \"in
fix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rsp
ace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \+
\"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"
false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+
\"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ
esetting:-mi(\"theta\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"
));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6
#Q!F'-F#6&-F,6#Q&omegaF'-I#moGF$63Q\"=F'/%%formGQ&infixF'/%&fenceGQ&fa
lseF'/%*separatorGF=/%'lspaceGQ/thickmathspaceF'/%'rspaceGFB/%)stretch
yGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopG
F=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%s
izeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6
&F+-I&mfracGF$6*-F#6#-F563Q+&PartialD;F'/F9Q'prefixF'F;F>/FAQ$0emF'/FD
FhoFEFGFIFLFOFQFSFUFXFenFhn-F#6%F+-F#6$Fbo-F,6#Q\"tF'F+/%.linethicknes
sGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFfp/%)bevelledGF=FenFhn-F5
63Q1&InvisibleTimes;F'F8F;F>FgoFioFEFGFIFLFOFQFSFUFXFenFhn-F,6#Q&theta
F'F+F+" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 
0 "" }}{PARA 213 "" 0 "" {TEXT 200 50 "then we obtain the first order \+
system of equations" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 214 "" 0
 "" {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin
g:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\")
, Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mo(\"&PartialD;\",
 form = \"prefix\", fence = \"false\", separator = \"false\", lspace =
 \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false
\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova
blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm
ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25
5,255]\")), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow
(Typesetting:-mo(\"&PartialD;\", form = \"prefix\", fence = \"false\",
 separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy =
 \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+
\"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals
e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[
0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"t\")), Ty
pesetting:-mi(\"\")), linethickness = \"1\", denomalign = \"center\", \+
numalign = \"center\", bevelled = \"false\", foreground = \"[0,0,0]\",
 background = \"[255,255,255]\"), Typesetting:-mo(\"&InvisibleTimes;\"
, form = \"infix\", fence = \"false\", separator = \"false\", lspace =
 \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false
\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova
blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm
ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25
5,255]\"), Typesetting:-mi(\"theta\")), Typesetting:-mo(\"=\", form = \+
\"infix\", fence = \"false\", separator = \"false\", lspace = \"thickm
athspace\", rspace = \"thickmathspace\", stretchy = \"false\", symmetr
ic = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"f
alse\", movablelimits = \"false\", accent = \"false\", font_style_name
 = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background
 = \"[255,255,255]\"), Typesetting:-mi(\"omega\")), Typesetting:-mi(\"
\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF
$6#Q!F'-F#6&F+-F#6&F+-I&mfracGF$6*-F#6#-I#moGF$63Q+&PartialD;F'/%%form
GQ'prefixF'/%&fenceGQ&falseF'/%*separatorGFA/%'lspaceGQ$0emF'/%'rspace
GFF/%)stretchyGFA/%*symmetricGFA/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1
F'/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%0font_style_nameGQ+2D~
CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,2
55,255]F'-F#6%F+-F#6$F8-F,6#Q\"tF'F+/%.linethicknessGQ\"1F'/%+denomali
gnGQ'centerF'/%)numalignGF[p/%)bevelledGFAFinF\\o-F963Q1&InvisibleTime
s;F'/F=Q&infixF'F?FBFDFGFIFKFMFPFSFUFWFYFfnFinF\\o-F,6#Q&thetaF'-F963Q
\"=F'FcpF?FB/FEQ/thickmathspaceF'/FHF\\qFIFKFMFPFSFUFWFYFfnFinF\\o-F,6
#Q&omegaF'F+" }}{PARA 215 "" 0 "" {TEXT 200 0 "" }}{PARA 216 "" 0 "" 
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m
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pesetting:-mfrac(Typesetting:-mrow(Typesetting:-mo(\"&PartialD;\", for
m = \"prefix\", fence = \"false\", separator = \"false\", lspace = \"0
em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", m
axsize = \"infinity\", minsize = \"1\", largeop = \"false\", movableli
mits = \"false\", accent = \"false\", font_style_name = \"2D Comment\"
, size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255
]\")), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Type
setting:-mo(\"&PartialD;\", form = \"prefix\", fence = \"false\", sepa
rator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"fa
lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\",
 largeop = \"false\", movablelimits = \"false\", accent = \"false\", f
ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]
\", background = \"[255,255,255]\"), Typesetting:-mi(\"t\")), Typesett
ing:-mi(\"\")), linethickness = \"1\", denomalign = \"center\", numali
gn = \"center\", bevelled = \"false\", foreground = \"[0,0,0]\", backg
round = \"[255,255,255]\"), Typesetting:-mo(\"&InvisibleTimes;\", form
 = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em
\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", max
size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi
ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+
size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]
\"), Typesetting:-mi(\"omega\")), Typesetting:-mo(\"=\", form = \"infi
x\", fence = \"false\", separator = \"false\", lspace = \"thickmathspa
ce\", rspace = \"thickmathspace\", stretchy = \"false\", symmetric = \+
\"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false
\", movablelimits = \"false\", accent = \"false\", font_style_name = \+
\"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+
\"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", f
orm = \"prefix\", fence = \"false\", separator = \"false\", lspace = \+
\"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\"
, maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl
elimits = \"false\", accent = \"false\", font_style_name = \"2D Commen
t\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,
255]\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(\"g\"), \+
Typesetting:-mo(\"&InvisibleTimes;\", form = \"infix\", fence = \"fals
e\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretc
hy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize
 = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"f
alse\", font_style_name = \"2D Comment\", size = \"12\", foreground = \+
\"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typese
tting:-mi(\"sin\"), Typesetting:-mo(\"&ApplyFunction;\", form = \"infi
x\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspac
e = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"
infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fa
lse\", accent = \"false\", font_style_name = \"2D Comment\", size = \"
12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Types
etting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true
\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thin
mathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"i
nfinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fal
se\", accent = \"false\", font_style_name = \"2D Comment\", size = \"1
2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese
tting:-mrow(Typesetting:-mi(\"theta\")), Typesetting:-mo(\")\", form =
 \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thin
mathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symm
etric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+
\"false\", movablelimits = \"false\", accent = \"false\", font_style_n
ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro
und = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"
\")), Typesetting:-mrow(Typesetting:-mi(\"R\")), linethickness = \"1\"
, denomalign = \"center\", numalign = \"center\", bevelled = \"false\"
, foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetti
ng:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mr
owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F
+-F#6&F+-I&mfracGF$6*-F#6#-I#moGF$63Q+&PartialD;F'/%%formGQ'prefixF'/%
&fenceGQ&falseF'/%*separatorGFA/%'lspaceGQ$0emF'/%'rspaceGFF/%)stretch
yGFA/%*symmetricGFA/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopG
FA/%.movablelimitsGFA/%'accentGFA/%0font_style_nameGQ+2D~CommentF'/%%s
izeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6
%F+-F#6$F8-F,6#Q\"tF'F+/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'
/%)numalignGF[p/%)bevelledGFAFinF\\o-F963Q1&InvisibleTimes;F'/F=Q&infi
xF'F?FBFDFGFIFKFMFPFSFUFWFYFfnFinF\\o-F,6#Q&omegaF'-F963Q\"=F'FcpF?FB/
FEQ/thickmathspaceF'/FHF\\qFIFKFMFPFSFUFWFYFfnFinF\\o-F#6%-F963Q*&umin
us0;F'F<F?FBFDFGFIFKFMFPFSFUFWFYFfnFinF\\o-F46*-F#6&-F,6#Q\"gF'F`p-F#6
&-F,6#Q$sinF'-F963Q0&ApplyFunction;F'FcpF?FBFDFGFIFKFMFPFSFUFWFYFfnFin
F\\o-F#6%-F963Q\"(F'F</F@Q%trueF'FB/FEQ.thinmathspaceF'/FHFjr/FJFhrFKF
MFPFSFUFWFYFfnFinF\\o-F#6#-F,6#Q&thetaF'-F963Q\")F'/F=Q(postfixF'FgrFB
Fir/FHQ2verythinmathspaceF'F\\sFKFMFPFSFUFWFYFfnFinF\\oF+F+-F#6#-F,6#Q
\"RF'FfoFioF\\pF^pFinF\\oF+F+F+" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}
{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 217 "" 0 "" {TEXT 200 7 "where \+
 " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"theta\"));" "-I
%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q&theta
F'" }{TEXT 200 6 "(0)= n" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesettin
g:-mi(\"Pi\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF
'6#-I#miGF$6#Q#PiF'" }{TEXT 200 9 "/50, and " }{XPPEDIT 18 0 "Typesett
ing:-mrow(Typesetting:-mi(\"omega\"));" "-I%mrowG6#/I+modulenameG6\"I,
TypesettingGI(_syslibGF'6#-I#miGF$6#Q&omegaF'" }{TEXT 200 8 "(0) = 0."
 }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 218 "" 0 "" {TEXT 200 78 "We
 wish to apply the method of Picard Iteration  to obtain a solution of
 this " }}{PARA 219 "" 0 "" {TEXT 200 77 "non-linear system.  We will \+
need to use a power series approximation for sin(" }{XPPEDIT 18 0 "Typ
esetting:-mrow(Typesetting:-mi(\"theta\"));" "-I%mrowG6#/I+modulenameG
6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q&thetaF'" }{TEXT 200 2 ")." 
}}{PARA 220 "" 0 "" {TEXT 200 16 "Recall that sin(" }{XPPEDIT 18 0 "Ty
pesetting:-mrow(Typesetting:-mi(\"theta\"));" "-I%mrowG6#/I+modulename
G6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q&thetaF'" }{TEXT 200 19 ") \+
is approximately " }}{PARA 221 "" 0 "" {TEXT 200 0 "" }}{PARA 222 "" 0
 "" {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin
g:-mrow(Typesetting:-mi(\"theta\"), Typesetting:-mo(\"&minus;\", form \+
= \"infix\", fence = \"false\", separator = \"false\", lspace = \"medi
ummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", sym
metric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop =
 \"false\", movablelimits = \"false\", accent = \"false\", font_style_
name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgr
ound = \"[255,255,255]\"), Typesetting:-mfrac(Typesetting:-mrow(Typese
tting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"theta\"), Typeset
ting:-mn(\"3\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Ty
pesetting:-mrow(Typesetting:-mn(\"6\")), linethickness = \"1\", denoma
lign = \"center\", numalign = \"center\", bevelled = \"false\", foregr
ound = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mo(
\"+\", form = \"infix\", fence = \"false\", separator = \"false\", lsp
ace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"
false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1
\", largeop = \"false\", movablelimits = \"false\", accent = \"false\"
, font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0
,0]\", background = \"[255,255,255]\"), Typesetting:-mfrac(Typesetting
:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"thet
a\"), Typesetting:-mn(\"5\"), superscriptshift = \"0\"), Typesetting:-
mi(\"\")), Typesetting:-mrow(Typesetting:-mn(\"120\")), linethickness \+
= \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"
false\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Ty
pesetting:-mo(\"&minus;\", form = \"infix\", fence = \"false\", separa
tor = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathsp
ace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini
ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\",
 accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+
foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting
:-mfrac(Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typ
esetting:-mi(\"theta\"), Typesetting:-mn(\"7\"), superscriptshift = \"
0\"), Typesetting:-mi(\"\")), Typesetting:-mrow(Typesetting:-mi(\"\"),
 Typesetting:-mrow(Typesetting:-mn(\"7\"), Typesetting:-mo(\"!\", form
 = \"infix\", fence = \"false\", separator = \"false\", lspace = \"ver
ythinmathspace\", rspace = \"verythinmathspace\", stretchy = \"false\"
, symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", larg
eop = \"false\", movablelimits = \"false\", accent = \"false\", font_s
tyle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", b
ackground = \"[255,255,255]\")), Typesetting:-mi(\"\")), linethickness
 = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \+
\"false\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), \+
Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen
ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6*-F,6#Q&thetaF'-I
#moGF$63Q(&minus;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/
%'lspaceGQ0mediummathspaceF'/%'rspaceGFB/%)stretchyGF=/%*symmetricGF=/
%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF
=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregro
undGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-I&mfracGF$6*-F#6%F+-I%m
supGF$6%F1-I#mnGF$6#Q\"3F'/%1superscriptshiftGQ\"0F'F+-F#6#-Fdo6#Q\"6F
'/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFdp/%)beve
lledGF=FenFhn-F563Q\"+F'F8F;F>F@FCFEFGFIFLFOFQFSFUFXFenFhn-F\\o6*-F#6%
F+-Fao6%F1-Fdo6#Q\"5F'FgoF+-F#6#-Fdo6#Q$120F'F_pFbpFepFgpFenFhnF4-F\\o
6*-F#6%F+-Fao6%F1-Fdo6#Q\"7F'FgoF+-F#6%F+-F#6$F`r-F563Q\"!F'F8F;F>/FAQ
2verythinmathspaceF'/FDF[sFEFGFIFLFOFQFSFUFXFenFhnF+F_pFbpFepFgpFenFhn
F+F+" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 0 "
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 223 "" 0 "" {TEXT 
200 27 "We will begin by using sin(" }{XPPEDIT 18 0 "Typesetting:-mrow
(Typesetting:-mi(\"theta\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti
ngGI(_syslibGF'6#-I#miGF$6#Q&thetaF'" }{TEXT 200 30 ") in the differnt
ial equation." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" 
{TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f1 := (x,y,t) -> y;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f2 := (x,y,t) -> -sin(x);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x_0 := Pi/50.0; y_0 := 0.0
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "yEst := t -> y_0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xEst := t -> x_0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "xEst(1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "yEstNew := unapply(y_0 + int(f2(xEst(t),yEst
(t),t),t),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xEstNew := unapply
(x_0 + int(f1(xEst(t),yEst(t),t),t),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "xEst := xEstNew:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "yEst :=
 yEstNew:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 224 "" 0 "" {TEXT 200 33 "We no
w take an approximation for " }{XPPEDIT 18 0 "Typesetting:-mrow(Typese
tting:-mi(\"theta\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s
yslibGF'6#-I#miGF$6#Q&thetaF'" }}{PARA 225 "" 0 "" {TEXT 200 6 "with  \+
" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"theta\"));" "-I%
mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q&thetaF
'" }{TEXT 200 6 "(0) = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting
:-mi(\"Pi\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'
6#-I#miGF$6#Q#PiF'" }{TEXT 200 4 "/50." }}{PARA 0 "" 0 "" {TEXT 204 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f1 := (x,y,t) -> y;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f2 := (x,y,t) -> -x + x^3/6  ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x_0 := Pi/50.0; y_0 := 0.0;"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "yEst := t -> y_0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xEst := t -> x_0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "xEst(1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "yEstNew := unapply(y_0 + int(f2(xEst(t),yEst
(t),t),t),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xEstNew := unapply
(x_0 + int(f1(xEst(t),yEst(t),t),t),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "xEst := xEstNew;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "yEst :=
 yEstNew;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([xEst(t),(Pi/50)*
cos(t)],t=0..1.5, color= [red,blue]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ":" }}}{EXCHG {PARA 226 "" 0 "" {TEXT 200 33 "We now ta
ke an approximation for " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesettin
g:-mi(\"theta\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysli
bGF'6#-I#miGF$6#Q&thetaF'" }}{PARA 227 "" 0 "" {TEXT 200 6 "with  " }
{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"theta\"));" "-I%mro
wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q&thetaF'" 
}{TEXT 200 7 "(0) = n" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-
mi(\"Pi\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#
-I#miGF$6#Q#PiF'" }{TEXT 200 4 "/50." }}{PARA 0 "" 0 "" {TEXT 204 0 ""
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart; n :=49;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f1 := (x,y,t) -> y;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f2 := (x,y,t) -> -x + x^3/6 -x^5/12
0 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x_0 := n*Pi/50.0; y_
0 := 0.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "yEst := t -> y
_0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xEst := t -> x_0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "yEstNew := unapply(y_0 + int
(f2(xEst(t),yEst(t),t),t),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xE
stNew := unapply(x_0 + int(f1(xEst(t),yEst(t),t),t),t);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "xEst := xEstNew;" }}{PARA 0 "> " 0 "" {MPLTEXT
 1 0 16 "yEst := yEstNew;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot([
xEst(t),x_0*cos(t)],t=0..20, view=[0.0..3,-2..4], color=[red,blue], th
ickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot([yEst(t),-x_0*sin(t)],t=0..20.
0, color=[red,blue], thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "d2 := unapply(diff(diff(xEst(t),t),t),t):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "errxEst := t -> d2(t) + sin(xEst(t)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf((n*Pi/50)*errxE
st(0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "errHsol := t -> \+
-cos(t) + sin(cos(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e
valf(n*(Pi/50)*errHsol(0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 0 "" 
}}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 228 "" 0 "" {TEXT 200 55 "Let
 us see how the solution changes as a function of n." }}{PARA 0 "" 0 "
" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart;
 n :=25;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f1 := (x,y,t) -
> y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f2 := (x,y,t) -> -x
 + x^3/6 - x^5/120 + x^7/7! ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "x_0 := Pi/50.0; y_0 := 0.0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "yEst := (t,v) -> y_0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "xEst := (t,v) -> v*x_0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "yEstNew := unapply(y_0 + int(f2(xEst(t,n),yEst(t,n),t
),t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xEstNew := unap
ply(n*x_0 + int(f1(xEst(t,n),yEst(t,n),t),t),t);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "xEst := xEstNew;" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 16 "yEst := yEstNew;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "plot([xEst(t,n),n*x_0*cos(t)],t=0..1.9, color=[red,bl
ue], thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([yEst(t,n),-n*x_0*sin(t
)],t=0..2.0, color=[red,blue], thickness=2);" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 44 "D1 :=  unapply(diff(diff(xEst(t,v),t),t),t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "error1 := t -> D1(t) + sin(x
Est(t,n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(error1(
1.8));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }