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"" 1 "" {XPPMATH 20 "6#>%+eulersolveGf*6'%\"xG%\"yG%\" fG%\"hG%#x2G6\"6$%)operatorG%&arrowGF,@%19(9$9%6$-%&printG6&F3F49&9'-F $6$-%-oneeulerstepGF8F2F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eulersolve(0,0,(x,y)->5-y,.01,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF(,&\"\"&\"\"\"9%! \"\"F(F(F($F.!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"\"!\"#$\"\"& F%f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF,,&F'F$9%!\"\"F,F,F,F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"#!\"#$\"$&**!\"%f*6$%\"xG%\"yG6\" 6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"$!\"#$\"'0&[\"!\"'f*6$%\"xG%\"yG6\"6$%)operat orG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"%!\"#$\")&*>q>!\")f*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"&!\"#$\"+0v\\]C!#5f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-, &F$\"\"\"9%!\"\"F-F-F-$F2F%" }}{PARA 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"6&$\"#Y!\"#$\"+V=)3&=!\"* f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#Z!\"#$\"+DIP#)=!\"*f*6$%\"xG% \"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#[!\"#$\"+&H\\N\">!\"*f*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#\\!\"#$\"+-QTW>!\"*f*6$%\"xG%\"yG6\"6$%) operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&$\"#]!\"#$\"+k'p\\(>!\"*f*6$%\"xG%\"yG6\"6$%)operator G%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#^!\"#$\"+n*>_+#!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#_!\"#$\"+nx;N?!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,& \"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#` !\"#$\"+*3;[1#!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\" 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\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#p !\"#$\"+\\[\"3]#!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\" \"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#q!\"#$\" +,n!e_#!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\" \"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#r!\"#$\"+M'[0b#! \"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$ F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#s!\"#$\"+[J/vD!\"*f*6$%\"x G%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#t!\"#$\"+b2$!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\" \"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#'*!\"#$ \"+5Ww%4$!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%! \"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#(*!\"#$\"+mn\"Q 6$!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F- F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#)*!\"#$\"+)fyE8$!\"*f*6 $%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"#**!\"#$\"+7=N^J!\"*f*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF-,&\"\"&\"\"\"9%!\"\"F-F-F-$F3F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%HQ)pJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve(\{D(y)(t)=5-y(t),y(0)=0\});evalf(subs(t=1,5-5* exp(-t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&\"\"&\" \"\"*&F)F*-%$expG6#,$F'!\"\"F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+%z-1;$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# Runge-Ku tta method" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rksetup:=(x,y ,f,h) -> (x,y,x+h,y+f(x,y)*h,f,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%(rksetupGf*6&%\"xG%\"yG%\"fG%\"hG6\"6$%)operatorG%&arrowGF+6(9$9%,&F 0\"\"\"9'F3,&F1F3*&F4F3-9&6$F0F1F3F3F8F4F+F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "onerkstep:=(x,y,x2,y2,f,h)->rksetup(x+h,y+h*(f (x,y)+f(x2,y2))/2,f,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*onerkste pGf*6(%\"xG%\"yG%#x2G%#y2G%\"fG%\"hG6\"6$%)operatorG%&arrowGF--%(rkset upG6&,&9$\"\"\"9)F6,&9%F6*&#F6\"\"#F6*&F7F6,&-9(6$F5F9F6-F@6$9&9'F6F6F 6F6F@F7F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "prerksolve :=(x,y,x2,y2,f,h,x3) ->if x>=x3 then y else " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(print(x,y,x2,y2,f,h),prerksolve(onerkstep(x,y,x2,y2, f,h),x3)) end if;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+prerksolveGf*6 )%\"xG%\"yG%#x2G%#y2G%\"fG%\"hG%#x3G6\"6$%)operatorG%&arrowGF.@%19*9$9 %6$-%&printG6(F5F69&9'9(9)-F$6$-%*onerkstepGF:F4F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rksolve:=(x,y,f,h,x3)->prerksolve(r ksetup(x,y,f,h),x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rksolveGf*6 '%\"xG%\"yG%\"fG%\"hG%#x3G6\"6$%)operatorG%&arrowGF,-%+prerksolveG6$-% (rksetupG6&9$9%9&9'9(F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rksolve(0,0,(x,y)->5-y,.01,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( \"\"!F#$\"\"\"!\"#$\"\"&F&f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF-,&F( F%9%!\"\"F-F-F-F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"\"!\"#$\"+++ +v\\!#6$\"\"#F%$\"+++DD**F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,& \"\"&F$9%!\"\"F1F1F1F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"#!\"#$ \"+]()\\+**!#6$\"\"$F%$\"+w$\\,[\"!#5f*6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF2,&\"\"&\"\"\"9%!\"\"F2F2F2$F8F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"$!\"#$\"+z))px9!#5$\"\"%F%$\"+!*=#H'>F(f*6$%\"xG%\"yG6\"6$% )operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6($\"\"%!\"#$\"+vd\\g>!#5$\"\"&F%$\"+<3*3W#F(f*6$%\"xG% \"yG6\"6$%)operatorG%&arrowGF1,&F*\"\"\"9%!\"\"F1F1F1$F6F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"&!\"#$\"+U))[QC!#5$\"\"'F%$\"+aR59HF(f* 6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&F$\"\"\"9%!\"\"F1F1F1$F6F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"'!\"#$\"+yes6H!#5$\"\"(F%$\"+>'3E Q$F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1 $F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"(!\"#$\"+1UD!Q$!#5$\"\") F%$\"+k;XYQF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%! \"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\")!\"#$\"+x17WQ !#5$\"\"*F%$\"+q%zcI%F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"& \"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"*!\"# $\"+w;P.V!#5$\"#5F%$\"+fzLgZF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1 ,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\" #5!\"#$\"+GJ0eZ!#5$\"#6F%$\"+(fs/@&F(f*6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#6!\"#$\"+*\\5#3_!#5$\"#7F%$\"+%RGhl&F(f*6$%\"xG%\"yG6\"6$%)o peratorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#7!\"#$\"+0)))Ql&!#5$\"#8F%$\"+<*\\t4'F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#8!\"#$\"+6E8&4'!#5$\"#9F%$\"+&G\"=MlF(f* 6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#9!\"#$\"+Ug)>`'!#5$\"#:F%$\"+# =mm'pF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F 1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#:!\"#$\"+\"y#\\kp!#5$ \"#;F%$\"+`y%[R(F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\" \"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#;!\"#$\"+y gp#R(!#5$\"#jjB)!#5$\"#>F%$\"+k&**Rl)F(f*6$%\"xG%\"yG6\"6$%)o peratorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#>!\"#$\"+#Q6>l)!#5$\"#?F%$\"+oARl!*F(f*6$%\"xG%\"yG 6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#?!\"#$\"+k[Kj!*!#5$\"#@F%$\"+:;ps%*F(f*6$% \"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6($\"#@!\"#$\"+#yW1Z*!#5$\"#AF%$\"+M$Qf( )*F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1 $F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#A!\"#$\"+m=\"R()*!#5$\"#B F%$\"+vs^F5!\"*f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF2,&\"\"&\"\"\"9% !\"\"F2F2F2$F8F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#B!\"#$\"+WmJF5 !\"*$\"#CF%$\"+yM/n5F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"& \"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#C!\"#$ \"+V[%o1\"!\"*$\"#DF%$\"+&Rwh5\"F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arro wGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ($\"#D!\"#$\"+P(zf5\"!\"*$\"#EF%$\"+S*>\\9\"F(f*6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#E!\"#$\"+R_sW6!\"*$\"#FF%$\"+()zF$=\"F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#F!\"#$\"+B_3$=\"!\"*$\"#GF%$\"+rVD@7F(f* 6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#G!\"#$\"+DN1@7!\"*$\"#HF%$\"+! *G&)e7F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1 F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#H!\"#$\"+VRme7!\"*$\" #IF%$\"+/t2'H\"F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\" 9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#I!\"#$\"+P-* eH\"!\"*$\"#JF%$\"+N8$HL\"F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,& \"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#J !\"#$\"+HhuK8!\"*$\"#KF%$\"+o'=%p8F(f*6$%\"xG%\"yG6\"6$%)operatorG%&ar rowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#K!\"#$\"+0`Bp8!\"*$\"#LF%$\"+_Ha09F(f*6$%\"xG%\"yG6\"6$%)operat orG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#L!\"#$\"+99O09!\"*$\"#MF%$\"++yIT9F(f*6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6($\"#M!\"#$\"+o!G6W\"!\"*$\"#NF%$\"+(y;nZ\"F(f*6$ %\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#N!\"#$\"+W)QlZ\"!\"*$\"#OF%$\"+cM x6:F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F 1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#O!\"#$\"+#G(f6:!\"*$\"#P F%$\"+48[Y:F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%! \"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#P!\"#$\"+*)oIY: !\"*$\"#QF%$\"+?Q%3e\"F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\" &\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#Q!\"# $\"+N6n!e\"!\"*$\"#RF%$\"+CW'[h\"F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arr owGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6($\"#R!\"#$\"+dMp9;!\"*$\"#SF%$\"+Ala[;F(f*6$%\"xG%\"yG6\"6$%)operato rG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#S!\"#$\"+dsP[;!\"*$\"#TF%$\"+%[$*=o\"F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#T!\"#$\"+.fs\"o\"!\"*$\"#UF%$\"+W'3\\r\" F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F 7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#U!\"#$\"+IFu9\"=F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,& \"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#X !\"#$\"+]^$=\"=!\"*$\"#YF%$\"+)z;P%=F(f*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#Y!\"#$\"+!RdN%=!\"*$\"#ZF%$\"+;=7v=F(f*6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#Z!\"#$\"+%*R'\\(=!\"*$\"#[F%$\"+aV@1>F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#[!\"#$\"+-\"eg!>!\"*$\"#\\F%$\"+@v*p$>F( f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#\\!\"#$\"+CG%o$>!\"*$\"#]F%$ \"+'Ruu'>F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\" \"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#]!\"#$\"+Q7Kn>!\" *$\"#^F%$\"+E![w*>F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\" \"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#^!\"#$\" +#R'\\(*>!\"*$\"#_F%$\"+G9_F?F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF 1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($ \"#_!\"#$\"+.8PF?!\"*$\"#`F%$\"+!f(4d?F(f*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#`!\"#$\"+f*[p0#!\"*$\"#aF%$\"+p%zj3#F(f*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#a!\"#$\"+XeT#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\" \"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#m! \"#$\"+'o9dT#!\"*$\"#nF%$\"+RvbTCF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arr owGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6($\"#n!\"#$\"+D$G9W#!\"*$\"#oF%$\"+US,nCF(f*6$%\"xG%\"yG6\"6$%)operat orG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#o!\"#$\"+8h)oY#!\"*$\"#pF%$\"+_s@#\\#F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#p!\"#$\"+'f!4#\\#!\"*$\"#qF%$\"+!ppr^#F( f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#q!\"#$\"+&HWq^#!\"*$\"#rF%$\" +_Q(=a#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F 1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#r!\"#$\"+/(\\GF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F 1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#$)!\"#$\"+\"=B(>G!\"*$ \"#%)F%$\"+\\f_TGF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\" \"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#%)!\"#$\"+ NpTTG!\"*$\"#&)F%$\"+mF+jGF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,& \"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#& )!\"#$\"+O[*G'G!\"*$\"#')F%$\"+))eE%)GF(f*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#')!\"#$\"+K!fT)G!\"*$\"#()F%$\"+UuJ0HF(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#()!\"#$\"+];@0H!\"*$\"#))F%$\"+M&fh#HF(f *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#))!\"#$\"+%zag#H!\"*$\"#*)F%$ \"+YUzYHF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\" F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#*)!\"#$\"+\\0pYH!\" *$\"#!*F%$\"+WOAnHF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\" \"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#!*!\"#$ \"+y47nH!\"*$\"#\"*F%$\"+o(\\u)HF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arro wGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ($\"#\"*!\"#$\"+C\"[t)H!\"*$\"##*F%$\"+VYZ2IF(f*6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"##*!\"#$\"+5SP2I!\"*$\"#$*F%$\"+q-IFIF(f*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#$*!\"#$\"+R1?FI!\"*$\"#%*F%$\"+L'Gp/$F(f *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#%*!\"#$\"+$**Ho/$!\"*$\"#&*F%$ \"+$phj1$F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\" \"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#&*!\"#$\"+NSEmI! \"*$\"#'*F%$\"+&R,c3$F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"& \"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#'*!\"# $\"+3Z]&3$!\"*$\"#(*F%$\"+h'\\Y5$F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arr owGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6($\"#(*!\"#$\"+ORb/J!\"*$\"#)*F%$\"+(R3N7$F(f*6$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#)*!\"#$\"+COTBJ!\"*$\"#**F%$\"+)[z@9$F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#**!\"#$\"+ec3UJ!\"*$\"$+\"F%$\"+,[mgJF(f *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&\"\"&\"\"\"9%!\"\"F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1>dgJ!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "rksolve(0,2,(x,y)->x+y,.01,.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"!\"\"#$\"\"\"!\"#$\"$-#F'f*6$%\"xG%\"yG6\"6 $%)operatorG%&arrowGF.,&9$F&9%F&F.F.F.F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"\"!\"#$\"+++:??!\"*$\"\"#F%$\"++:XS?F(f*6$%\"xG%\"yG6\"6$%) operatorG%&arrowGF1,&9$F$9%F$F1F1F1F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"#!\"#$\"+3IgS?!\"*$\"\"$F%$\"+Q!471#F(f*6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"$!\"#$\"+o?Oh?!\"*$\"\"%F%$\"+*ovA3#F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"%!\"#$\"+d-V#3#!\"*$\"\"&F%$\"+gXl.@F(f*6$%\" 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\"*$\"#7F%$\"+*R;BE#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\" \"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#7!\"#$\"+TQ[i A!\"*$\"#8F%$\"+z'3jG#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\" \"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#8!\"#$\"+/y Z'G#!\"*$\"#9F%$\"+#eU1J#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9 $\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#9!\"#$\" +1M\"3J#!\"*$\"#:F%$\"+S:KNBF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1 ,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#:!\"# $\"+\"3%\\NB!\"*$\"#;F%$\"+A!\\.O#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&ar rowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\" #;!\"#$\"+(HB0O#!\"*$\"#F%$\"+6ybPCF(f*6$%\"xG%\"yG6\"6$%)operat orG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#>!\"#$\"+$RPxV#!\"*$\"#?F%$\"+nZ,kCF(f*6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#?!\"#$\"+ah>kC!\"*$\"#@F%$\"+;\"Q3\\#F(f*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#@!\"#$\"+D8-\"\\#!\"*$\"#AF%$\"+Q:.=DF(f*6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#A!\"#$\"+*e;#=D!\"*$\"#BF%$\"+b()fXDF(f* 6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6($\"#B!\"#$\"+mcyXD!\"*$\"#CF%$\"+BNatD F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#C!\"#$\"+7BttD!\"*$\"#DF%$\"+N'p= g#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#D!\"#$\"+A.1-E!\"*$\"#EF%$\"+D 4eIEF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7F %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#E!\"#$\"+GNxIE!\"*$\"#FF%$\"+ j7ofEF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F1$F7 F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#F!\"#$\"+-e()fE!\"*$\"#GF%$ \"+gX<*o#F(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F1F 1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#G!\"#$\"+a5P*o#!\"*$\"#H F%$\"+lZ1>FF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F1F 1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"#H!\"#$\"+LKE>F!\"*$\"# IF%$\"+leN\\FF(f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,&9$\"\"\"9%F7F 1F1F1$F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gjb\\F!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dsolve(\{D(y)(t)=y(t)+t,y(0) =2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(F'!\"\"\"\" \"F)*&\"\"$F*-%$expGF&F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(-.3-1+3*exp(.3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ckd \\F!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# fourth order Runge Kutta" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s1:=(stepsize,tvalue,xvalue, f)->f(tvalue,xvalue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1Gf*6&%)s tepsizeG%'tvalueG%'xvalueG%\"fG6\"6$%)operatorG%&arrowGF+-9'6$9%9&F+F+ F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "s2:=(stepsize,tvalue, xvalue,f)->f(tvalue+stepsize/2,xvalue+(stepsize/2)*f(tvalue,xvalue)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s2Gf*6&%)stepsizeG%'tvalueG%'xv alueG%\"fG6\"6$%)operatorG%&arrowGF+-9'6$,&9%\"\"\"*&#F4\"\"#F49$F4F4, &9&F4*&F6F4*&F8F4-F06$F3F:F4F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "s3:=(stepsize,tvalue,xvalue,f)->f(tvalue+stepsize/2, xvalue+stepsize/2*s2(stepsize,tvalue,xvalue,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s3Gf*6&%)stepsizeG%'tvalueG%'xvalueG%\"fG6\"6$%)oper atorG%&arrowGF+-9'6$,&9%\"\"\"*&#F4\"\"#F49$F4F4,&9&F4*&F6F4*&F8F4-%#s 2G6&F8F3F:F0F4F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "s4:=(stepsize,tvalue,xvalue,f)->f(tvalue+stepsize,xvalue+stepsize*s3( stepsize,tvalue,xvalue,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s4Gf *6&%)stepsizeG%'tvalueG%'xvalueG%\"fG6\"6$%)operatorG%&arrowGF+-9'6$,& 9$\"\"\"9%F4,&9&F4*&F3F4-%#s3G6&F3F5F7F0F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "s5:=(stepsize,tvalue,xvalue,f)->(1/6)*steps ize*(s1+2*s2+2*s3+s4)(stepsize,tvalue,xvalue,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s5Gf*6&%)stepsizeG%'tvalueG%'xvalueG%\"fG6\"6$%)oper atorG%&arrowGF+,$*&#\"\"\"\"\"'F2*&9$F2-,*%#s1GF2*&\"\"#F2%#s2GF2F2*&F :F2%#s3GF2F2%#s4GF26&F59%9&9'F2F2F2F+F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 106 "onerk4step:=(stepsize,tvalue,xvalue,f)-> (stepsize ,tvalue+stepsize,xvalue+s5(stepsize,tvalue,xvalue,f),f);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%+onerk4stepGf*6&%)stepsizeG%'tvalueG%'xvalueG% \"fG6\"6$%)operatorG%&arrowGF+6&9$,&F0\"\"\"9%F2,&9&F2-%#s5G6&F0F3F59' F2F9F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "rk4solve:=(st epsize,tvalue,xvalue,f,goal)->if tvalue>=goal then xvalue" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 89 "else (print(tvalue,xvalue),rk4solve((onerk4s 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