{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 128 0 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 128 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 24 "Euler's Method-Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 77 "App roximate, using Euler's method, the solution of the differential equat ion " }}{PARA 259 "" 0 "" {TEXT -1 65 " dy/dx \+ = 2x-y where y(0) = 1 " }}{PARA 260 "" 0 "" {TEXT -1 39 "a nd compare it with the exact solution." }}{PARA 0 "" 0 "" {TEXT -1 19 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots) :" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "ExactSolution := (x) -> 3*exp(-x) +2*x -2;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%.ExactSolutionG:6#%\"xG6\"6$%)operatorG%&arrowGF(,(-%$expG6#,$9$!\" \"\"\"$F1\"\"#!\"#\"\"\"F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "picture1 := plot(ExactSolution(x), x=0..5, color=COLO R(RGB,0,0,1)):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h:=1; # the step size" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "oldx:=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\"!" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "oldy: =1; # the initial condition" }{XPPMATH 20 "6#>%%oldyG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lastx:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lastxG\"#5" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "L:=[]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=[op( L),[oldx,oldy]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7#7$\"\"!\" \"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "slope:=(x,y) -> 2*x- y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&slopeG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$\" \"#9%!\"\"F)F)" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "A While Loop " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "while oldx < lastx do ol dy:=oldy+h*slope(oldx,oldy);oldx:=oldx+h;L:=[op(L),[oldx,oldy]];od;" } }{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%o ldyG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7$7$\"\"!\"\"\"7$F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%oldxG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7%7$\"\"!\"\"\"7 $F(F'7$\"\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7&7$\"\"!\"\"\"7$F(F'7$\"\"#F+7$\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG 7'7$\"\"!\"\"\"7$F(F'7$\"\"#F+7$\"\"$\"\"%7$F.\"\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%oldyG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%o ldxG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7(7$\"\"!\"\"\"7$F( F'7$\"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldx G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7)7$\"\"!\"\"\"7$F(F'7 $\"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"&\"\")7$F0\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldx G\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7*7$\"\"!\"\"\"7$F(F'7 $\"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"&\"\")7$F0\"#57$\"\"(\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7+7$\"\"! \"\"\"7$F(F'7$\"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"&\"\")7$F0\"#57$\"\"( \"#77$F3\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldxG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7,7$\"\"!\"\"\"7$F(F'7$\"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"& \"\")7$F0\"#57$\"\"(\"#77$F3\"#97$\"\"*\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldx G\"#5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG7-7$\"\"!\"\"\"7$F(F'7$ \"\"#F+7$\"\"$\"\"%7$F.\"\"'7$\"\"&\"\")7$F0\"#57$\"\"(\"#77$F3\"#97$ \"\"*\"#;7$F5\"#=" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Plot for A pproximation" }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "picture2:=plot(L,style=point,symbol=diamond );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)picture2G-%/INTERFACE_PLOTG6%-%'C URVESG6$7-7$\"\"!$\"\"\"F-7$F.F-7$$\"\"#F-F27$$\"\"$F-$\"\"%F-7$F7$\" \"'F-7$$\"\"&F-$\"\")F-7$F:$\"#5F-7$$\"\"(F-$\"#7F-7$F?$\"#9F-7$$\"\"* F-$\"#;F-7$FB$\"#=F--%'COLOURG6&%$RGBG$FC!\"\"F-F--%'SYMBOLG6#%(DIAMON DG-%&STYLEG6#%&POINTG" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "Plot t he Exact and Approximation Solutions together." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(\{picture1,picture2\});" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6&7-7$\"\"!$\"\"\"F(7$F)F(7$$\"\"#F(F-7$$\" \"$F($\"\"%F(7$F2$\"\"'F(7$$\"\"&F($\"\")F(7$F5$\"#5F(7$$\"\"(F($\"#7F (7$F:$\"#9F(7$$\"\"*F($\"#;F(7$F=$\"#=F(-%'COLOURG6&%$RGBG$F>!\"\"F(F( -%'SYMBOLG6#%(DIAMONDG-%&STYLEG6#%&POINTG-F$6$7SF'7$$\"1LLL3x&)*3\"!#; 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This is a special case where the true solut ion approaches a straight line (y=2x-2) as x increases; this means Eul er's method approximation approaches the true solution as x increases \+ regardless of the step size h." }}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 }