{VERSION 6 1 "Windows XP" "6.1" } {USTYLETAB {PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 -1 3 3 2 0 2 0 2 2 -1 3 }{PSTYLE "W arning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 -1 3 3 2 0 2 0 2 2 -1 2 }{PSTYLE "_pstyle4" -1 200 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle3" -1 201 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 202 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 203 1 {CSTYLE "" -1 -1 "Cour ier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "MS Serif" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }1 1 0 -1 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3 " -1 5 1 {CSTYLE "" -1 -1 "MS Serif" 1 14 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }1 1 0 -1 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 1 0 -1 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head ing 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 1 0 -1 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 1 0 -1 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Left Justifie d Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 -1 3 3 2 0 2 0 2 2 -1 5 }{PSTYLE "E rror" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }3 1 0 -1 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined " -1 44 "Times" 1 12 0 0 0 0 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "Output Labe ls" -1 29 "Times" 1 8 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Maple Inpu t" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Page Num ber" -1 33 "Times" 1 10 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Maple In put Placeholder" -1 200 "Courier" 1 12 200 0 200 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small" -1 202 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 0 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Serif" 1 12 0 128 128 1 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "" 1 10 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 0 0 0 2 1 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "" 0 1 104 64 92 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Tim es" 1 12 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 " MS Serif" 1 12 147 0 15 1 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 " " 0 1 0 128 128 1 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 0 1 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "" 1 8 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 203 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 204 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 205 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 206 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle5" -1 204 1 {CSTYLE "" -1 -1 "Cou rier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle6" -1 205 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cst yle4" -1 207 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_psty le7" -1 206 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle8" -1 207 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle5" -1 208 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle9" -1 208 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_ pstyle10" -1 209 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle6" -1 209 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle11" -1 210 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 20 "#Andrew Sommese 20 04" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 28 "f:= (x,y,w) -> -w^2-y+ln(x);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 16 "initialValue:=0;" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 23 "endValue:=evalf(ln(2));" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 5 "a:=1;" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 5 "b:=2;" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6%I\"xGF%I\"yGF%I\"wGF%F%6$I)operatorGF%I&ar rowGF%F%,(*$9&\"\"#!\"\"9%F2-I#lnGF%6#9$\"\"\"F%F%F%" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I-initialValueG6\"\"\"!" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I)endValueG6\"$\"+1=ZJp!# 5" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"\"\"\"" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"bG6\"\"\"#" } {TEXT 207 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 61 "ic := \{ y(1)=initialValue, D(y)(1)=t, yt(1)=0, D(yt)(1)=1\};" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#icG6\"<&/--I\"DG6$I*prote ctedGF,I(_syslibGF%6#I\"yGF%6#\"\"\"I\"tGF%/-I#ytGF%F0\"\"!/--F*6#F5F0 F1/-F/F0F6" }{TEXT 207 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 47 "sys :=\{diff(y(x),x$2)-f(x,y(x),diff(y(x),x))=0," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 63 "\n diff(yt(x),x$2)+yt(x)+2*diff(y(x),x)*d iff(yt(x),x)=0\}; " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 37 "\n \+ fcns := \{y(x),yt(x)\};" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I$sysG6\"<$/,*-I %diffGI*protectedGF+6$-I\"yGF%6#I\"xGF%-I\"$GF+6$F0\"\"#\"\"\"*$-F*6$F -F0F4F5F-F5-I#lnG6$F+I(_syslibGF%F/!\"\"\"\"!/,(-F*6$-I#ytGF%F/F1F5FCF 5*&F7F5-F*6$FCF0F5F4F>" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I%fcnsG6\"<$-I\"yGF%6#I\"xGF%-I#ytGF%F)" }{TEXT 207 0 "" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 8 "t:= 0.6;" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"tG6\"$\"\"'!\"\"" }{TEXT 207 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 58 "p:=dsolve(sys union ic,fcns,numeric,met hod=classical[rk4]," }{MPLTEXT 1 -1 14 "stepsize=0.05," }{MPLTEXT 1 0 22 "output=listprocedure):" }{MPLTEXT 1 -1 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 17 "yy:=subs(p,y(x)):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\nyyt:= subs(p,yt(x)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> \+ " 0 "" {MPLTEXT 1 0 6 "n:=20;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 19 "\n h:=evalf((b-a)/n);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 20 "\nxx:=array(0 ..n,[]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\nsol:=array(0..n,[]);" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"nG6\"\"#?" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"$\"+++++]! #6" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#xxG6\"-I&arr ayGI*protectedGF(6$;\"\"!\"#?7\"" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I$solG6\"-I&arrayGI*protectedGF(6$;\"\"!\"#?7\"" } {TEXT 207 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 20 "for j fr om 0 to n do" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 15 "xx[j] := a+j*h;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 20 "\nsol[j]:= y y(xx[j]);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 3 "od :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 19 "w ith(CurveFitting):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 66 "\nsltn:= unap ply(Spline(convert(xx,'list'),convert(sol,'list'),u)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 22 "plot(\{sltn(u), ln(u)\}," }{MPLTEXT 1 0 1 "u" } {MPLTEXT 1 0 7 "=1..2);" }{MPLTEXT 1 0 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F* 7$$\"3hmm;arz@5!#<$\"3'>e7Jtw]H\"!#>7$$\"3OL$e9ui2/\"F/$\"3)>n\\%4')>: CF27$$\"3smm\"z_\"4i5F/$\"31svY$e^.m$F27$$\"3qmmT&phN3\"F/$\"3K7'4!4/_ '*[F27$$\"3UL$e*=)H\\5\"F/$\"3iS?t$okD6'F27$$\"3sm;z/3uC6F/$\"3!4]=Xi \">HsF27$$\"3-+]7LRDX6F/$\"3MMnS#>TYP)F27$$\"3em;zR'ok;\"F/$\"3i#Q,j@$ zZ&*F27$$\"3-+]i5`h(=\"F/$\"3Ye24lcgq5!#=7$$\"3YLL$3En$47F/$\"33]z**>; m)=\"Fen7$$\"3cmmT!RE&G7F/$\"3'fg#*=l$y\"H\"Fen7$$\"3)*****\\K]4]7F/$ \"3p87Z\"*=&pS\"Fen7$$\"3))****\\PAvr7F/$\"3P;xm>Yl@:Fen7$$\"3/++]nHi# H\"F/$\"3eyVxClMJ;Fen7$$\"3bm;z*ev:J\"F/$\"3(G$yX\\WFIFen7$$\"3#*** \\7o7Tv8F/$\"3],%QpO$3f?Fen7$$\"3ALLLQ*o]R\"F/$\"37+AL\"[\"3f@Fen7$$\" 3-+]7=lj;9F/$\"3[q'eHcs\"oAFen7$$\"3&***\\PaRTLowx9P#Fen7$$ \"3GLLe9Ege9F/$\"33w+]H*)pyCFen7$$\"3WL$eR\"3Gy9F/$\"3/@)*QyvnwDFen7$$ \"3mmmT5k]*\\\"F/$\"3Q(>OM;o=o#Fen7$$\"3em;zRQb@:F/$\"3kf:`V#>1z#Fen7$ $\"3%)**\\(=>Y2a\"F/$\"3u9/!HV%)[)GFen7$$\"3imm\"zXu9c\"F/$\"3gB\"**G` %H')HFen7$$\"3'******\\y))Ge\"F/$\"3g^)y6\")[14$Fen7$$\"3!****\\i_QQg \"F/$\"3%\\>er]bB>$Fen7$$\"3#***\\7y%3Ti\"F/$\"3S#*G)[f?/H$Fen7$$\"3#* ***\\P![hY;F/$\"35&*=\"*3K$*)R$Fen7$$\"3ELLLQx$om\"F/$\"3!H1,'\\87'\\$ Fen7$$\"3')****\\P+V)o\"F/$\"3MU,%GMz&*f$Fen7$$\"3im;zpe*zq\"F/$\"3'>> *)o@lIp$Fen7$$\"3)*****\\#\\'QHv(>S*=%Fen7$$\"3em mm'*RRL=F/$\"3?)H0*)H/rG%Fen7$$\"3_mmTvJga=F/$\"35l5'Go@oQ%Fen7$$\"3KL $e9tOc(=F/$\"3o8J3Nu_&[%Fen7$$\"3'******\\Qk\\*=F/$\"3CF/$\"3%*>YlW1pzYFen7$$\"3_mmmw(Gp$>F/$\"3_p_t29@sZFen7$$\" 3-+]7oK0e>F/$\"3]F/$\"3=8G&Q<0['\\Fen7$$\" \"#F*$\"3_8Y>qp'e1&Fen-%&COLORG6&%$RGBG$\"#5!\"\"$F*F_[lF`[l-F$6$7SF'7 $F-$\"3o+;Hz#*Hc@F27$F4$\"3Cvs9f\\Q&*RF27$F9$\"3V!)[]e.,CgF27$F>$\"3Nz g%G7[`-)F27$FC$\"3ifDyp?=y**F27$FH$\"3UnYqLh_v6Fen7$FM$\"3/'RE2)QEc8Fe n7$FR$\"3Z-#))4G4)R:Fen7$FW$\"3(>7m*eNZ>Fen7$F \\o$\"3aT,pXR:e?Fen7$Fao$\"3p)zVUu&>KAFen7$Ffo$\"3S9'e5kcRS#Fen7$F[p$ \"3aY)*R@YtmDFen7$F`p$\"3s7O*za\"H7FFen7$Fep$\"3i*z`2qfE)GFen7$Fjp$\"3 2ef'G\"RB%*f80%Fen7$Fbs$\"3'G**Rsv?t>%Fen7$Fgs$\"3&)HU_#R oEK%Fen7$F\\t$\"3c<90-aIcWFen7$Fat$\"3_A[(eA:Df%Fen7$Fft$\"3Fqhc\\$)*R s%Fen7$F[u$\"3qUn$QO!f\\[Fen7$F`u$\"3%*3MPja@()\\Fen7$Feu$\"36%e&)\\h# G4^Fen7$Fju$\"33Voz_7*zB&Fen7$F_v$\"3wCp3sn?``Fen7$Fdv$\"3QTP,qrmxaFen 7$Fiv$\"31c2hHpP$f&Fen7$F^w$\"3fNqi5E\"Hr&Fen7$Fcw$\"30')Rd\\_TGeFen7$ Fhw$\"3g#)HL'>Bz%fFen7$F]x$\"3cEz8@*)ohgFen7$Fbx$\"3I:km:vqwhFen7$Fgx$ \"3)=9e)=>[*G'Fen7$F\\y$\"3[e_iT/+#R'Fen7$Fay$\"3-i%QA.?#3lFen7$Ffy$\" 3N)4(HQh.6mFen7$F[z$\"3l@k8\"\\2&>nFen7$F`z$\"3V^^#)yyBAoFen7$Fez$\"3' GX*f0=ZJpFen-Fjz6&F\\[lF`[lF][lF`[l-%+AXESLABELSG6$Q\"u6\"Q!Fjdl-%*GRI DSTYLEG6#%,RECTANGULARG-%%VIEWG6$;F][l$\"#?F_[l;$!23*)>6O%H'Q\"Fen$\"1 W9rT75qq!#;-%*LINESTYLEG6#F*-%+PROJECTIONG6#F][l" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 208 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 8 "t:= 1.0;" }{MPLTEXT 1 0 0 "" }} {PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"tG6\"$\"#5!\"\"" }{TEXT 207 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 58 "p:=dsolve(sys union ic, fcns,numeric,method=classical[rk4]," }{MPLTEXT 1 -1 14 "stepsize=0.05, " }{MPLTEXT 1 0 22 "output=listprocedure):" }{MPLTEXT 1 -1 0 "" }} {PARA 204 "> " 0 "" {MPLTEXT 1 0 17 "yy:=subs(p,y(x)):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\nyyt:= subs(p,yt(x)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 6 "n:=20;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 19 "\nh:=evalf((b-a)/n);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 20 "\nxx:=array(0..n,[]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\n sol:=array(0..n,[]);" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I\"nG6\"\"#?" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 " 6#>I\"hG6\"$\"+++++]!#6" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#xxG6\"-I&arrayGI*protectedGF(6$;\"\"!\"#?7\"" }{TEXT 207 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I$solG6\"-I&arrayGI*protectedGF(6 $;\"\"!\"#?7\"" }{TEXT 207 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 20 "for j from 0 to n do" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 15 "xx[j] := a+j*h;" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 20 "\nsol[j]:= yy(xx[j]);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 3 "od:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 19 "with(CurveFitting):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 66 "\nsltn:= unapply(Spline(convert(xx,'list'),convert( sol,'list'),u)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{sltn(u), ln(u)\},u=1..2);" }{MPLTEXT 1 0 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%'CURVESG6$7 S7$$\"\"\"\"\"!$F*F*7$$\"3hmm;arz@5!#<$\"3YP7$$\"3OL$e9ui2/ \"F/$\"3yn\")>!Q55*RF27$$\"3smm\"z_\"4i5F/$\"3_9t[oo#p-'F27$$\"3qmmT&p hN3\"F/$\"3!yAW!>+_F!)F27$$\"3UL$e*=)H\\5\"F/$\"3s1')>sQvx**F27$$\"3sm ;z/3uC6F/$\"33a?)3+Wa<\"!#=7$$\"3-+]7LRDX6F/$\"3MwJ'\\q[iN\"FL7$$\"3em ;zR'ok;\"F/$\"3,KQRIE$)R:FL7$$\"3-+]i5`h(=\"F/$\"3EEHRk^[>FL7$$\"3cmmT!RE&G7F/$\"3-5XoB([\"e?FL7$$\"3)*** **\\K]4]7F/$\"3cJC;7a>KAFL7$$\"3))****\\PAvr7F/$\"3M^Q;&HeRS#FL7$$\"3/ ++]nHi#H\"F/$\"3)4?(4<[tmDFL7$$\"3bm;z*ev:J\"F/$\"3Vu'yW\"4H7FFL7$$\"3 ELL$347TL\"F/$\"3-R/B`#fE)GFL7$$\"3=LLLjM?`8F/$\"3/PA(=\"ouCIFL7$$\"3# ***\\7o7Tv8F/$\"3eA$)*o'y_(=$FL7$$\"3ALLLQ*o]R\"F/$\"3=&4vY&zVHLFL7$$ \"3-+]7=lj;9F/$\"3wn\"4T%Q&G[$FL7$$\"3&***\\PaR%FL7$$\"3 %)**\\(=>Y2a\"F/$\"3)\\gP***zmAVFL7$$\"3imm\"zXu9c\"F/$\"3-Xf^3]IcWFL7 $$\"3'******\\y))Ge\"F/$\"3wQ_\"*z[^#f%FL7$$\"3!****\\i_QQg\"F/$\"3%=% fo))y*Rs%FL7$$\"3#***\\7y%3Ti\"F/$\"31CcBa+f\\[FL7$$\"3#****\\P![hY;F/ $\"3ste*o+:s)\\FL7$$\"3ELLLQx$om\"F/$\"3)=BQt:#G4^FL7$$\"3')****\\P+V) o\"F/$\"3>_!*4S2*zB&FL7$$\"3im;zpe*zq\"F/$\"3Y^Gn[k?``FL7$$\"3)***** \\#\\'QHR'FL7$$\"3@LL3dg6<>F/$\"3LYfbKgF3lFL7$$ \"3_mmmw(Gp$>F/$\"3p#G+=;66h'FL7$$\"3-+]7oK0e>F/$\"3yCun\"*HT>nFL7$$\" 3-+](=5s#y>F/$\"3y0J^i_$>#oFL7$$\"\"#F*$\"38l9wJ8ZJpFL-%&COLORG6&%$RGB G$\"#5!\"\"$F*F_[lF`[l-F$6$7SF'7$F-$\"3o+;Hz#*Hc@F27$F4$\"3Cvs9f\\Q&*R F27$F9$\"3V!)[]e.,CgF27$F>$\"3Nzg%G7[`-)F27$FC$\"3ifDyp?=y**F27$FH$\"3 UnYqLh_v6FL7$FN$\"3/'RE2)QEc8FL7$FS$\"3Z-#))4G4)R:FL7$FX$\"3(>7m*eNZ>< FL7$Fgn$\"3Z^f/zH(4!>FL7$F\\o$\"3aT,pXR:e?FL7$Fao$\"3p)zVUu&>KAFL7$Ffo $\"3S9'e5kcRS#FL7$F[p$\"3aY)*R@YtmDFL7$F`p$\"3s7O*za\"H7FFL7$Fep$\"3i* z`2qfE)GFL7$Fjp$\"32ef'G\"RB%*f80%FL7$Fbs$\"3'G**Rsv?t>%FL7$Fgs$ \"3&)HU_#RoEK%FL7$F\\t$\"3c<90-aIcWFL7$Fat$\"3_A[(eA:Df%FL7$Fft$\"3Fqh c\\$)*Rs%FL7$F[u$\"3qUn$QO!f\\[FL7$F`u$\"3%*3MPja@()\\FL7$Feu$\"36%e&) \\h#G4^FL7$Fju$\"33Voz_7*zB&FL7$F_v$\"3wCp3sn?``FL7$Fdv$\"3QTP,qrmxaFL 7$Fiv$\"31c2hHpP$f&FL7$F^w$\"3fNqi5E\"Hr&FL7$Fcw$\"30')Rd\\_TGeFL7$Fhw $\"3g#)HL'>Bz%fFL7$F]x$\"3cEz8@*)ohgFL7$Fbx$\"3I:km:vqwhFL7$Fgx$\"3)=9 e)=>[*G'FL7$F\\y$\"3[e_iT/+#R'FL7$Fay$\"3-i%QA.?#3lFL7$Ffy$\"3N)4(HQh. 6mFL7$F[z$\"3l@k8\"\\2&>nFL7$F`z$\"3V^^#)yyBAoFL7$Fez$\"3'GX*f0=ZJpFL- Fjz6&F\\[lF`[lF][lF`[l-%+AXESLABELSG6$Q\"u6\"Q!Fjdl-%*GRIDSTYLEG6#%,RE CTANGULARG-%%VIEWG6$;F][l$\"#?F_[l;$!23*)>6O%H'Q\"FL$\"1W9rT75qq!#;-%* LINESTYLEG6#F*-%+PROJECTIONG6#F][l" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 208 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 17 "sltn := proc(u,t)" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 58 "\nlocal f,initialValue,endValue,a,b,ic,sys,fcns,p,yy, yyt,n," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 19 "\n h,xx,sol,j;" } {MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 28 "f:= (x,y,w) - > -w^2-y+ln(x);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 16 "initialValue:=0;" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 23 "endValue:=evalf(ln(2));" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 5 "a:=1;" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 5 "b:=2;" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 " " {MPLTEXT 1 0 61 "ic := \{y(1)=initialValue, D(y)(1)=t, yt(1)=0, D (yt)(1)=1\}:" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 47 "sys :=\{diff(y(x),x$2)-f(x,y(x),diff(y(x),x))=0," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 63 "\n diff(yt(x),x$2)+yt(x)+2*diff(y(x),x)*dif f(yt(x),x)=0\}: " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 37 "\n \+ fcns := \{y(x),yt(x)\}:" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 206 "> " 0 "" {MPLTEXT 1 0 58 "p:=dsolve(sys union ic,fcns,numeric,method=classical[rk4]," }{MPLTEXT 1 -1 14 "step size=0.05," }{MPLTEXT 1 0 22 "output=listprocedure):" }{MPLTEXT 1 -1 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 17 "yy:=subs(p,y(x)):" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\nyyt:= subs(p,yt(x)):" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 6 "n:=20;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 19 "\nh:=evalf((b-a)/n);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 20 "\nxx:=array(0..n,[]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\nsol:=array(0..n,[]);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 20 "for j from 0 to n do" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 15 "xx[j] := a+j*h;" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 20 "\nsol[j]:= yy(xx[j]);" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 3 "od:" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 61 "unapply(Spline(convert(xx,'list'),convert(sol,' list'),u))(u):" }{MPLTEXT 1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 25 "plot(sltn(u,0.3),u=1..2);" }{MPLTEXT 1 0 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F*7$$\"3hmm;arz@5!#<$\"3 c0c+gdB3l!#?7$$\"3OL$e9ui2/\"F/$\"3hdw8h#[e@\"!#>7$$\"3smm\"z_\"4i5F/$ \"37T'))\\M%e[=F87$$\"3qmmT&phN3\"F/$\"3GPDQqfo#[#F87$$\"3UL$e*=)H\\5 \"F/$\"3$Hq]j'o:7JF87$$\"3sm;z/3uC6F/$\"3+WK8;$*=&p$F87$$\"3-+]7LRDX6F /$\"3WENE'fP()H%F87$$\"3em;zR'ok;\"F/$\"3*pFJ'[b@(R(F87$$\"3))**** \\PAvr7F/$\"32j[RJ&)oV!)F87$$\"3/++]nHi#H\"F/$\"3uxpnY=!)p')F87$$\"3bm ;z*ev:J\"F/$\"3#*oa6sGOT#*F87$$\"3ELL$347TL\"F/$\"3L8v_S;6D**F87$$\"3= LLLjM?`8F/$\"37]jL2W\"30\"!#=7$$\"3#***\\7o7Tv8F/$\"3h>-$3f6\">6F^q7$$ \"3ALLLQ*o]R\"F/$\"3Y9P&)RM-!=\"F^q7$$\"3-+]7=lj;9F/$\"3)zTm'z*ztC\"F^ q7$$\"3&***\\PaRY2a\"F/ $\"3!y%GM3F=Z;F^q7$$\"3imm\"zXu9c\"F/$\"35pQW)=,ir\"F^q7$$\"3'****** \\y))Ge\"F/$\"3aq))Rk_B)y\"F^q7$$\"3!****\\i_QQg\"F/$\"3w9>[!=V%f=F^q7 $$\"3#***\\7y%3Ti\"F/$\"3rELU[%\\!H>F^q7$$\"3#****\\P![hY;F/$\"3-'3Wa+ or+#F^q7$$\"3ELLLQx$om\"F/$\"3yuMOo<7y?F^q7$$\"3')****\\P+V)o\"F/$\"3) ejIj(poa@F^q7$$\"3im;zpe*zq\"F/$\"3J==M\"\\&zCAF^q7$$\"3)*****\\#\\'QH rXd8!3$HF^q7$$\"3@LL3dg6<>F/$\"3+u#4)>Ky=IF^q7$$\"3 _mmmw(Gp$>F/$\"3_7SP4GH)4$F^q7$$\"3-+]7oK0e>F/$\"3Us$\\#RH-%=$F^q7$$\" 3-+](=5s#y>F/$\"3k!\\_4N'*oE$F^q7$$\"\"#F*$\"3Ae'o]H:kN$F^q-%&COLORG6& %$RGBG$\"#5!\"\"$F*F`[lFa[l-%+AXESLABELSG6$Q\"u6\"Q!Ff[l-%*GRIDSTYLEG6 #%,RECTANGULARG-%%VIEWG6$;F^[l$\"#?F`[l;$!1;t8!fIGr'F^q$\"2;.q4gVNU$F/ -%*LINESTYLEG6#F*-%+PROJECTIONG6#F^[l" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}{TEXT 208 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 14 "sltn(1.3,1.5);" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#$\"+_lp*p$!#5" }{TEXT 207 0 "" }}}{EXCHG {PARA 204 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 208 "" 0 "" {TEXT 203 0 "" }} {PARA 208 "" 0 "" {TEXT 203 0 "" }}{PARA 209 "" 0 "" {TEXT 209 0 "" }} {PARA 210 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }