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2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 259 "" 0 "" {TEXT 324 38 "Maple Tools for Differential E quations" }}{PARA 258 "" 0 "" {TEXT 325 0 "" }{TEXT -1 10 "A. J. Meir " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 46 "C opyright (C) A. J. Meir. All rights reserved." }}{PARA 257 "" 0 "" {TEXT -1 492 "This worksheet is for educational use only. No part of t his publication may be reproduced or transmitted for profit in any for m or by any means, electronic or mechanical, including photocopy, reco rding, or any information storage and retrieval system without prior w ritten permission from the author. Not for profit distribution of the \+ software is allowed without prior written permission, providing that t he worksheet is not modified in any way and full credit to the author \+ is acknowledged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 326 91 "(This document is largely based on a similar document or iginally written by P. G. Schmidt.)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Maple will find " }{TEXT 256 18 "symbolic solutions" }{TEXT -1 11 " (that is, " }{TEXT 257 21 " closed-form solutions" }{TEXT -1 4 " or " }{TEXT 258 17 "solution form ulas" }{TEXT -1 131 ") for many types of d.e.s and i.v.p.s, including \+ separable and linear first-order equations. Moreover, Maple can comput e and graph " }{TEXT 259 19 "numerical solutions" }{TEXT -1 11 " (that is, " }{TEXT 260 21 "approximate solutions" }{TEXT -1 50 ") for virtu ally all i.v.p.s of practical interest." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 43 "1. Finding and Plotting Symbolic So lutions." }{TEXT -1 60 " To find a general solution for a given d.e., use Maple's " }{TEXT 262 6 "dsolve" }{TEXT -1 34 " command, whose g eneral format is" }}{PARA 256 "" 0 "" {TEXT 263 27 "dsolve(DE,unknown_ function)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "The followin g command, for example, finds a general solution for the d.e. " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-k*x" " 6#,$*&%\"kG\"\"\"%\"xGF&!\"\"" }{TEXT -1 67 " (the radioactive-decay \+ equation), where the unknown function is " }{XPPEDIT 18 0 "x=x(t)" "6 #/%\"xG-F$6#%\"tG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 17 " is a parameter:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(D(x)(t)=-k*x(t),x(t)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&%$_C1G\"\"\"-%$ex pG6#,$*&%\"kGF*F'F*!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Note that inside the " }{TEXT 264 6 "dsolve" } {TEXT -1 27 " command the derivative, " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 26 "', should be written as " }{XPPEDIT 18 0 "D(x)" "6#- %\"DG6#%\"xG" }{TEXT -1 38 " and that the independent variable, " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 52 ", must be carried along (so that the d.e. becomes " }{XPPEDIT 18 0 "D(x)(t)=-k*x(t)" "6#/--%\"DG 6#%\"xG6#%\"tG,$*&%\"kG\"\"\"-F(6#F*F.!\"\"" }{TEXT -1 14 " instead o f " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "- k*x" "6#,$*&%\"kG\"\"\"%\"xGF&!\"\"" }{TEXT -1 26 " ). In Maple's resp onse, " }{XPPEDIT 18 0 "_C1" "6#%$_C1G" }{TEXT -1 84 " denotes an ar bitrary constant. (In general, Maple renders arbitrary constants as \+ " }{XPPEDIT 18 0 "_C1" "6#%$_C1G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "_C 2" "6#%$_C2G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "_C3" "6#%$_C3G" } {TEXT -1 87 ", and so forth.) In standard calculus notation, we would write the solution formula as" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "x(t)=c*exp(-k*t)" "6#/-%\"xG6#%\"tG*&%\"cG\"\"\"-%$expG 6#,$*&%\"kGF*F'F*!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "The next command produces a general solution for the d.e. " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 "'' + " }{XPPEDIT 18 0 "a^2*x= 0" "6#/*&%\"aG\"\"#%\"xG\"\"\"\"\"!" }{TEXT -1 37 " (the elastic-spri ng equation with " }{XPPEDIT 18 0 "a^2=k/m" "6#/*$%\"aG\"\"#*&%\"kG\" \"\"%\"mG!\"\"" }{TEXT -1 41 " ), where the unknown function is again \+ " }{XPPEDIT 18 0 "x=x(t)" "6#/%\"xG-F$6#%\"tG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 17 " is a parameter:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ds olve(D(D(x))(t)+a^2*x(t),x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"xG6#%\"tG,&*&%$_C1G\"\"\"-%$sinG6#*&%\"aGF+F'F+F+F+*&%$_C2GF+-%$cosG F.F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Note that the second derivative, " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 27 "'', should be written as " }{XPPEDIT 18 0 "D(D(x))" "6# -%\"DG6#-F$6#%\"xG" }{TEXT -1 91 " and that the independent variable \+ must be carried along, just as in the previous example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "With a minor modific ation, we can use the " }{TEXT 287 6 "dsolve" }{TEXT -1 72 " command to find symbolic solutions of i.v.p.s; the modified format is" }} {PARA 256 "" 0 "" {TEXT 265 50 "dsolve(\{DE,initial_condition(s)\},unk nown_function)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 115 "Note t he \"curly braces\" around the d.e. and initial condition(s). For exam ple, let's solve the first-order i.v.p. " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-k*x" "6#,$*&%\"kG\"\"\"%\"xGF&! \"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x(0)=x0" "6#/-%\"xG6#\"\"!%#x 0G" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dsolve(\{D(x)(t)=-k*x(t),x(0)=x0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&%#x0G\"\"\"-%$expG6# ,$*&%\"kGF*F'F*!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Next, we solve the second-order i.v.p. " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 "'' + " }{XPPEDIT 18 0 "a^2*x=0" "6#/*& %\"aG\"\"#%\"xG\"\"\"\"\"!" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "x(0)=x0 " "6#/-%\"xG6#\"\"!%#x0G" }{TEXT -1 12 ", x'(0) = " }{XPPEDIT 18 0 " v0" "6#%#v0G" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dsolve(\{D(D(x))(t)+a^2*x(t) =0,x(0)=x0,D(x)(0)=v0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"xG6#%\"tG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF,F'F,F,F,F1!\"\"F,*&%#x0G F,-%$cosGF/F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "In order to " }{TEXT 266 4 "plot" }{TEXT -1 30 " a soluti on generated by the " }{TEXT 267 6 "dsolve" }{TEXT -1 50 " command, \+ we first have to give it a name, say, " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 30 ". The following command sets " }{XPPEDIT 18 0 "s" "6#% \"sG" }{TEXT -1 77 " equal to the right-hand side of the immediately \+ preceding solution formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "s:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF)%\"tGF)F)F)F.! \"\"F)*&%#x0GF)-%$cosGF,F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "Note: The percent (%) is just a placehol der for the " }{TEXT 268 61 "last output generated before execution of the present command" }{TEXT -1 84 " (which, in this case, happens to \+ be the solution formula produced by the preceding " }{TEXT 269 6 "dsol ve" }{TEXT -1 74 " command). Instead of using the percent, we could ha ve given a name (say, " }{XPPEDIT 18 0 "sol" "6#%$solG" }{TEXT -1 53 " ) to the solution formula and used that name in the " }{TEXT 270 3 "r hs" }{TEXT -1 9 " command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol:=dsolve(\{D(D(x))(t)+a^2*x(t)=0 ,x(0)=x0,D(x)(0)=v0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sol G/-%\"xG6#%\"tG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF.F)F.F.F.F3!\"\"F.*&% #x0GF.-%$cosGF1F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s:=r hs(sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*&*&%#v0G\"\"\"-%$ sinG6#*&%\"aGF)%\"tGF)F)F)F.!\"\"F)*&%#x0GF)-%$cosGF,F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "If we now specify the value of the parameter " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 26 " and the initial values " }{XPPEDIT 18 0 "x0" "6#%#x0G" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v0" "6#%#v0G" }{TEXT -1 16 ", we can p lot " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 51 " just like any othe r function. Set, for example, " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\" " }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x0=2" "6#/%#x0G\"\"#" }{TEXT -1 9 " , and " }{XPPEDIT 18 0 "v0=-3" "6#/%#v0G,$\"\"$!\"\"" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=1; x0:=2; v0:=-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "In this case, the solution of t he i.v.p. is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#% \"tG!\"$*&\"\"#\"\"\"-%$cosGF&F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 45 "and a graph can be generated with the us ual " }{TEXT 271 4 "plot" }{TEXT -1 10 " command:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(s,t=- 10..20);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6%-%'C URVESG6$7cy7$$!#5\"\"!$!3k95#3R1-J$!#<7$$!3J++vV8_O)*F-$!3gKL:W=ZLIF-7 $$!3i++](oUIn*F-$!3K#)ftJt%en#F-7$$!3%4+]7.k&4&*F-$!3O:ht)>poC#F-7$$!3 [*****\\P&3Y$*F-$!3k,Z)>\\wzv\"F-7$$!3;*\\7.ZVQ?*F-$!3U7D&fphRH\"F-7$$ !3k+]il:gh!*F-$!3x-$zE'f5Q!)!#=7$$!3K+v$4mf$>*)F-$!3MmfBEfBuHFL7$$!3-+ +Dcx6x()F-$\"3OE8U;(4(\\@FL7$$!3$**\\7y&=:<')F-$\"3]taXpE8byFL7$$!3%)* *\\Pff=d%)F-$\"3p#>2+#y*fL\"F-7$$!3u*\\P41?sH)F-$\"3a'Gb`CoB&=F-7$$!3n ****\\iTDP\")F-$\"3zga/Z#R9K#F-7$$!3A***\\i!zAwzF-$\"3R(Q3;jFPt#F-7$$! 3n******\\;?:yF-$\"3-q08\")\\GvIF-7$$!37++v$RvTl(F-$\"3gI(z)>SFPLF-7$$ !3m****\\P\"\\J\\(F-$\"3FZq'3=;H^$F-7$$!3Q\\(=Un=IT(F-$\"3a&e\"[dUkmNF -7$$!3)**\\P4@))GL(F-$\"3M,`p]N[(f$F-7$$!3g]ilZxv_sF-$\"3z9Rp]hB0OF-7$ $!3I+]P%GFE<(F-$\"3k&G1bI_)*e$F-7$$!3t*\\7yNmB,(F-$\"3y!fWP-=-\\$F-7$$ !3/++DJa5_oF-$\"3-2_m'>M6I$F-7$$!36++]PI_.nF-$\"31D=?t(3,0$F-7$$!3=++v V1%\\b'F-$\"39KRK=2(=t#F-7$$!3D+++]#ejS'F-$\"3y_=9rFV`BF-7$$!3K++Dcexd iF-$\"362#oTCMJ#>F-7$$!3E++v$RFR5'F-$\"3m9JitU0L9F-7$$!3?++DJ*y+&fF-$ \"3AMr$G#p@\"4*FL7$$!39++vo/B'z&F-$\"3V\">Z>-Krj$FL7$$!33++D1?QUcF-$!3 4cXbc<(G!>FL7$$!3/++D1/`d8B=F-7$$!3$****\\i!3%f+&F-$!3g.'\\J oeHH#F-7$$!3E*****\\xSt%[F-$!3-51&fOQOq#F-7$$!3[***\\PuS()o%F-$!3Iu?]X KXYIF-7$$!3s****\\729IXF-$!3O$3QqA)z7LF-7$$!3%****\\7oS:P%F-$!3#R%*o'3 x)f\\$F-7$$!3g\\(oz'3(**G%F-$!3y`nI/_BcNF-7$$!39+voa5S3UF-$!3q8T!y#R$G f$F-7$$!3p]iST7$o7%F-$!3e#RD'*\\Sbg$F-7$$!3M+]7G9EXSF-$!3a))Qv@/F%f$F- 7$$!33+Dc,=7#)QF-$!36*p8O/f-]$F-7$$!3$)*****\\<#)*=PF-$!3'[=#R=jH8LF-7 $$!3e**\\7.()GvNF-$!3;JQTG,_vIF-7$$!3x***\\7B&fJMF-$!3)*emBi.NuFF-7$$! 3'***\\Pf)HR.# FL7$$!3w****\\()*>ZL#F-$\"3?i*4)=uhGyFL7$$!3G+++]4Hs@F-$\"3YOj#3[@')4?F-$\"3+6&4O3g_'=F-7$$!3'3++](GVZ=F-$\"3h7*eBF&pRBF-7$ $!3e++++C!4p\"F-$\"3H$e9?w_(QFF-7$$!3I+++D>PM:F-$\"3#y#Rd5G%32$F-7$$!3 -+++]9%yP\"F-$\"3DYC#y@XyK$F-7$$!3u*****\\(4J@7F-$\"3=Wkq6eZ.NF-7$$!3w \\iST'Q-:\"F-$\"3MUp#3SB^b$F-7$$!3y*\\7yIm\"z5F-$\"3Ec&>ij?))e$F-7$$!3 !)\\(=U(R435F-$\"34t`YvtR/OF-7$$!3=)**\\iS;-P*FL$\"3qw3Ep\\x,OF-7$$!3c )*\\PM(p([zFL$\"3\"*e$pv]x?a$F-7$$!3%*)***\\iIKFlFL$\"30=\"R^TK4T$F-7$ $!3'))*\\P%[&3P[FL$\"3I7]#='*[d;$F-7$$!3w)**\\i!z%o9$FL$\"3g$='>X#Q.$G F-7$$!3o)*\\7G.hc9FL$\"3mHH!G$)fUT#F-7$$\"3(R,++]siL#!#>$\"3%zv<'3CPH> F-7$$\"3;****\\(=![l;FL$\"3)GUYEV))\\Z\"F-7$$\"3#p****\\7Lt4$FL$\"3#4* ppA$eT!**FL7$$\"3n%***\\ig=HXFL$\"3eeF-$!3(zW')e?$)R^$F-7$$\"3 S*\\P%)*\\#Q,#F-$!3)4U21'ysnNF-7$$\"3[uoa5UEa?F-$!3cW\\bj$pee$F-7$$\"3 _\\ilAMq%4#F-$!33i\"*H'eZ\")f$F-7$$\"3gCcwME9N@F-$!3mar@=CF-$!3q'y)))=t!\\[$F-7$$\"3? 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For further details on pl otting, especially on simultaneous plotting of several graphs, see the handout \"" }{HYPERLNK 17 "More on Basic Maple Commands" 1 "more.mws " "" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 53 "2. Plotting Direction Fields and Numerical Solutions. " }{TEXT -1 115 " The basic Maple command for the graphical represent ation of direction fields and approximate solution curves is " } {TEXT 274 6 "DEplot" }{TEXT -1 48 " ; it is part of a package of rout ines called " }{TEXT 275 7 "DEtools" }{TEXT -1 51 ", which must be e xplicitly loaded with the command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "To plot a " } {TEXT 276 15 "direction field" }{TEXT -1 37 " for a first-order d.e. o f the form " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " } {XPPEDIT 18 0 "f(t,x)" "6#-%\"fG6$%\"tG%\"xG" }{TEXT -1 21 ", in the \+ rectangle " }{XPPEDIT 18 0 "t=a..b" "6#/%\"tG;%\"aG%\"bG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "x=c..d" "6#/%\"xG;%\"cG%\"dG" }{TEXT -1 12 ", \+ use the " }{TEXT 277 6 "DEplot" }{TEXT -1 21 " command in the form " }}{PARA 256 "" 0 "" {TEXT 278 44 "DEplot(D(x)(t)=f(t,x(t)),x(t),t=a. .b,x=c..d)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 "The followi ng command, for example, generates a direction field for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG \"\"#" }{TEXT -1 20 " in the rectangle " }{XPPEDIT 18 0 "t=-3..3" "6 #/%\"tG;,$\"\"$!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "x=-6..6" "6# /%\"xG;,$\"\"'!\"\"F'" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "DEplot(D(x)(t)=x(t)^2,x( t),t=-3..3,x=-6..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6%-%'CURVESG6]dl7%7$$!+C%e(3I!\"*$!+qKI:jF*7$$!+w:C\"*HF* $!+Inp%o&F*7$$!+%*pWMIF*$!+Ry%o\"eF*7%7$$!+]*oHp#F*F+7$$!+-@XvEF*F07$$ !+?vl=FF*F57%7$$!+w%zrP#F*F+7$$!+GEmfBF*F07$$!+Y!oGS#F*F57%7$$!+-+Rh?F *F+7$$!+aJ(Q/#F*F07$$!+s&yq3#F*F57%7$$!+G0gXFapF57%7$$\"+YJO\"\\\"F apF+7$$\"+I;`m;FapF07$$\"+auZM7FapF57%7$$\"+$)yD\\YFapF+7$$\"+njUC[Fap F07$$\"+\">sBR%FapF57%7$$\"+?E:2yFapF+7$$\"+/6K#)zFapF07$$\"+GpE]vFapF 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initial points do not have to \+ lie inside the plot window." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "If initial conditions are given, specification \+ of the vertical range (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 35 "-r ange) is optional; if omitted, " }{TEXT 285 6 "DEplot" }{TEXT -1 18 " will choose the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 182 "-rang e automatically, depending on where the solution curves \"go\". This i s sometimes useful, but not always. 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\\dlFi`w7$F_dlF\\aw7$FbdlF_aw7%7$FfdlFi`w7$FidlF\\aw7$F\\elF_aw7%7$F($ \"+Enp%o&F*7$F.$\"+kKI:jF*7$F3$\"+b@:$='F*7%7$F9F_fw7$F(\\UiiDp(F`\\x7$Fh_x$!3]D>KNbC]iF`\\x7$F]`x$!3u)4QPQf LE&F`\\x7$$\"+++++7F*$!3/;0,#*fhXXF`\\x7$$\"+++++:F*$!3'*G'RW6I,+%F`\\ x7$$\"+++++=F*$!3!)[W$Q$[`rNF`\\x7$$\"+++++@F*$!3xZO%*)\\%*eA$F`\\x7$$ \"+++++CF*$!3XOG*\\V]7%HF`\\x7$$\"+++++FF*$!3([')3rhlFq#F`\\x7$$Fc_xF* $!3I+9;:T0+DF`\\x-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-Fd[x6&%$RGBG$ \"\"*F`axFgdx$\"\"#F`ax-%%VIEWG6$;$!+++++LF*$\"+++++LF*;$!+++++mF*$\"+ ++++mF*-%+AXESLABELSG6$Q\"t6\"Q%x(t)F\\fx" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Can you explain the po or outcome? (Hint: The " }{TEXT 286 5 "exact" }{TEXT -1 34 " solution \+ with initial condition " }{XPPEDIT 18 0 "x(0)=1" "6#/-%\"xG6#\"\"!\" \"\"" }{TEXT -1 31 " has a vertical asymptote at " }{XPPEDIT 18 0 "t =1" "6#/%\"tG\"\"\"" }{TEXT -1 3 " .)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 121 "If only the solution curves are of i nterest, display of the direction field arrows can be suppressed by ad ding the option" }}{PARA 256 "" 0 "" {TEXT 289 11 "arrows=none" }} {PARA 0 "" 0 "" {TEXT -1 12 "inside the " }{TEXT 290 6 "DEplot" } {TEXT -1 70 " command, following the specification of initial conditi ons. Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "DEplot(D(x)(t)=x(t)^2,x(t),t=-3..3,x=-6..6,\{[x( 0)=1],[x(0)=-1]\},arrows=none);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6&-%'CURVESG6&777$$!+++++I!\"*$\"339`+++++D!#=7$$!+++ ++FF*$\"3C\"*)3Fq-Fq#F-7$$!+++++CF*$\"3yEbfqk(\\UiiDp(F-7$Fao$!3]D>KNbC]iF-7$Ffo$!3u)4QPQfLE&F-7$ $\"+++++7F*$!3/;0,#*fhXXF-7$$\"+++++:F*$!3'*G'RW6I,+%F-7$$\"+++++=F*$! 3!)[W$Q$[`rNF-7$$\"+++++@F*$!3xZO%*)\\%*eA$F-7$$\"+++++CF*$!3XOG*\\V]7 %HF-7$$\"+++++FF*$!3([')3rhlFq#F-7$$F\\oF*$!3I+9;:T0+DF--%&STYLEG6#%%L INEG-%*THICKNESSG6#\"\"$-%&COLORG6&%$RGBG$\"\"*FipFat$\"\"#Fip-%%VIEWG 6$;$!+++++LF*$\"+++++LF*;$!+++++mF*$\"+++++mF*-%+AXESLABELSG6$Q\"t6\"Q %x(t)Ffu" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Many further options are available. In fact, most of the options avai lable with the basic " }{TEXT 291 4 "plot" }{TEXT -1 31 " command wi ll also work with " }{TEXT 292 6 "DEplot" }{TEXT 318 5 ". Ver" } {TEXT -1 44 "y useful for printing purposes is the option" }}{PARA 256 "" 0 "" {TEXT 293 15 "linecolor=black" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 340 "which causes the solution curves to be drawn in \+ black rather than yellow (the default, which looks pitiful on a black- and-white printout). The color and appearance of the direction field a rrows can be customized, too, but this will hardly ever be necessary. \+ For details, consult the relevant help pages (by typing ?DEplot or ?pl ot,options)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Note: Options may be added to the " }{TEXT 294 6 "DEplot" } {TEXT -1 118 " command in any order, but must be preceded by the rang e specification(s) and initial condition(s) (if any). Example:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "DEplot(D(x)(t)=x(t)^2,x(t),t=-3..3,x=-6..6,\{[x(0)=1],[x(0)=-1]\} ,\ndirgrid=[10,10],linecolor=black,color=green,title=`dx/dt=x^2`);" }} {PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6(-%'CURVESG6aq7%7 $$!+2+\\=I!\"*$!+O-klmF*7$$!+$**4:)HF*$!+k(fVL&F*7$$!+I9ssIF*$!+QaM8cF *7%7$$!+SL#=N#F*F+7$$!+EL%[J#F*F07$$!+jZ01CF*F57%7$$!+tm:&o\"F*F+7$$!+ fm<[;F*F07$$!+'4)QRhg#FRF57%7$$\"+X**4:)*FRF+7$$\"+3+\\=5F*F07$$\"+4dy s#*FRF57%7$$\"+hm<[;F*F+7$$\"+vm:&o\"F*F07$$\"+Q_%Rf\"F*F57%7$$\"+GL%[ J#F*F+7$$\"+UL#=N#F*F07$$\"+0>hgAF*F57%7$$\"+&**4:)HF*F+7$$\"+4+\\=IF* F07$$\"+s&ys#HF*F57%7$$!+kR[IIF*$!+k'R0L&F*7$$!+Og^pHF*$!+qOz-SF*7$$!+ /,qlIF*$!+Gr,vUF*7%7$$!+(Hr!\\JF*$!+&z%G'f#F*7$$!+-)G4& GF*$!+2_r.9F*7$$!+CT,!*HF*$!+)\\yte\"F*7%7$$!+JXS#[#F*Fd_l7$$!+N@E%=#F *Fi_l7$$!+duMBBF*F^`l7%7$$!+kyt:=F*Fd_l7$$!+oaf<:F*Fi_l7$$!+!z!oc;F*F^ `l7%7$$!+)>r!\\6F*Fd_l7$$!+4!)G4&)FRFi_l7$$!+D79+**FRF^`l7%7$$!+5`/C[F RFd_l7$$!+U8iU=FRFi_l7$$!+eXZLKFRF^`l7%7$$\"+d8iU=FRFd_l7$$\"+D`/C[FRF i_l7$$\"+4@>LMFRF^`l7%7$$\"+E!)G4&)FRFd_l7$$\"+*>r!\\6F*Fi_l7$$\"+xe)* 45F*F^`l7%7$$\"+qaf<:F*Fd_l7$$\"+myt:=F*Fi_l7$$\"+WDlw;F*F^`l7%7$$\"+P @E%=#F*Fd_l7$$\"+LXS#[#F*Fi_l7$$\"+6#>LM#F*F^`l7%7$$\"+/)G4&GF*Fd_l7$$ \"++72\\JF*Fi_l7$$\"+ye)*4IF*F^`l7%7$$!+(o&RDLF*$!+P(pG6)FR7$$!+8VguEF *$!+BOY?_FR7$$!+UFR7%7$$!+?!H(eEF*F^fl7$$!+Yw$z+#F*Fcfl 7$$!+]c\"p;#F*Fhfl7%7$$!+`B1#*>F*F^fl7$$!+z4FT8F*Fcfl7$$!+$)*[-]\"F*Fh fl7%7$$!+'o&RD8F*F^fl7$$!+FJ/YnFRFcfl7$$!+jJ#eL)FRFhfl7%7$$!+#>!H(e'FR F^fl7$$!)gkPzFRFcfl7$$!+'\\c\"p;FRFhfl7%7$$\")vkPzFRF^fl7$$\"+2-H(e'FR Fcfl7$$\"+r,^(*\\FRFhfl7%7$$\"+WJ/YnFRF^fl7$$\"+)o&RD8F*Fcfl7$$\"+%oF*Fcfl7$$\"+^V3L=F*Fhfl7%7$$ \"+[w$z+#F*F^fl7$$\"+A!H(eEF*Fcfl7$$\"+=5v*\\#F*Fhfl7%7$$\"+:VguEF*F^f l7$$\"+*o&RDLF*Fcfl7$$\"+&o!H(e'FRFf\\m7$$!)ekPzFRFi\\m7$$!+%\\c\"p;FRF\\]m7%7$$\")t kPzFRFf\\m7$$\"+4-H(e'FRFi\\m7$$\"+t,^(*\\FRF\\]m7%7$$\"+UJ/YnFRFf\\m7 $FajlFi\\m7$FdjlF\\]m7%7$FhjlFf\\m7$F[[mFi\\m7$F^[mF\\]m7%7$Fb[mFf\\m7 $Fe[mFi\\m7$Fh[mF\\]m7%7$F\\\\mFf\\m7$F_\\mFi\\m7$Fb\\mF\\]m7%7$$!+*>r !\\JF*$\"+/_r.9F*7$$!+,)G4&GF*$\"+#z%G'f#F*7$$!+BT,!*HF*$\"+-:i7CF*7%7 $$!+KXS#[#F*Fh`m7$$!+M@E%=#F*F]am7$$!+cuMBBF*Fbam7%7$$!+lyt:=F*Fh`m7$$ !+naf<:F*F]am7$$!+*y!oc;F*Fbam7%7$FfalFh`m7$$!+1!)G4&)FRF]am7$$!+C79+* *FRFbam7%7$$!+8`/C[FRFh`m7$$!+R8iU=FRF]am7$$!+dXZLKFRFbam7%7$$\"+a8iU= FRFh`m7$$\"+G`/C[FRF]am7$$\"+5@>LMFRFbam7%7$$\"+B!)G4&)FRFh`m7$$\"++72 \\6F*F]am7$$\"+ye)*45F*Fbam7%7$$\"+paf<:F*Fh`m7$$\"+nyt:=F*F]am7$$\"+X Dlw;F*Fbam7%7$$\"+O@E%=#F*Fh`m7$$\"+MXS#[#F*F]am7$$\"+7#>LM#F*Fbam7%7$ $\"+.)G4&GF*Fh`m7$$\"+,72\\JF*F]am7$$\"+ze)*4IF*Fbam7%7$Fhx$\"+:6@xEF* 7$F]y$\"+ZbX*)RF*7$Fby$\"+$zr[t$F*7%7$FhyF^gm7$F[zFagm7$F^zFdgm7%7$Fbz F^gm7$FezFagm7$FhzFdgm7%7$F\\[lF^gm7$$!+)***[4%*FRFagm7$Fb[lFdgm7%7$$! +@L%Q#RFRF^gm7$$!+JL#Gu#FRFagm7$$!+HWn=QFRFdgm7%7$$\"+YL#Gu#FRF^gm7$$ \"+OL%Q#RFRFagm7$$\"+QA*z%GFRFdgm7%7$$\"+:+\\4%*FRF^gm7$F]]lFagm7$F`]l Fdgm7%7$Fd]lF^gm7$Fg]lFagm7$Fj]lFdgm7%7$F^^lF^gm7$Fa^lFagm7$Fd^lFdgm7% 7$Fh^lF^gm7$F[_lFagm7$F^_lFdgm7%7$F^r$\"+nOz-SF*7$Fcr$\"+h'R0L&F*7$Fhr $\"+.iJe]F*7%7$F^sF\\[n7$FasF_[n7$FdsFb[n7%7$FhsF\\[n7$F[tF_[n7$F^tFb[ n7%7$$!+kR[I5F*F\\[n7$FetF_[n7$FhtFb[n7%7$F\\uF\\[n7$F_uF_[n7$$!+oVL!* RFRFb[n7%7$FfuF\\[n7$FiuF_[n7$$\"+*HKjn#FRFb[n7%7$F`vF\\[n7$FcvF_[n7$$ \"+m*)*HM*FRFb[n7%7$FjvF\\[n7$F]wF_[n7$F`wFb[n7%7$FdwF\\[n7$FgwF_[n7$F jwFb[n7%7$F^xF\\[n7$FaxF_[n7$FdxFb[n7%7$F($\"+i(fVL&F*7$F.$\"+K-klmF*7 $F3$\"+fXl'Q'F*7%7$F9Fb^n7$Fhg#FRFh^n7% 7$$\"+W**4:)*FRFb^n7$FioFe^n7$F\\pFh^n7%7$F`pFb^n7$FcpFe^n7$FfpFh^n7%7 $FjpFb^n7$F]qFe^n7$F`qFh^n7%7$FdqFb^n7$FgqFe^n7$FjqFh^n-%'COLOURG6&%$R GBG$\"\"!Fcbn$\"*++++\"!\")Fbbn-F$6&777$$!+++++IF*$\"339`+++++D!#=7$$! +++++FF*$\"3C\"*)3Fq-Fq#F_cn7$$!+++++CF*$\"3yEbfqk(\\UiiD p(F_cn7$Fffn$!3]D>KNbC]iF_cn7$F[gn$!3u)4QPQfLE&F_cn7$$\"+++++7F*$!3/;0 ,#*fhXXF_cn7$$\"+++++:F*$!3'*G'RW6I,+%F_cn7$$\"+++++=F*$!3!)[W$Q$[`rNF _cn7$$\"+++++@F*$!3xZO%*)\\%*eA$F_cn7$$\"+++++CF*$!3XOG*\\V]7%HF_cn7$$ \"+++++FF*$!3([')3rhlFq#F_cn7$$FafnF*$!3I+9;:T0+DF_cn-%&STYLEG6#%%LINE G-%*THICKNESSG6#\"\"$-F_bn6&FabnFcbnFcbnFcbn-%%VIEWG6$;$!+++++OF*$\"++ +++OF*;$!+++++sF*$\"+++++sF*-%+AXESLABELSG6$Q\"t6\"Q%x(t)Fe\\o-%&TITLE G6#%*dx/dt=x^2G" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 25 "The default method that " }{TEXT 295 6 "DEplot" } {TEXT -1 236 " uses to compute approximate solutions of the given d.e . is \"rk4\", a so-called 4th order Runge-Kutta method, much more sop histicated than the classical Euler method. You can force Maple to use the Euler method by specifying the option" }}{PARA 256 "" 0 "" {TEXT 296 16 "method=classical" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 74 "Both \"rk4\" and \"Euler\" employ a fixed stepsize; the default st epsize is (" }{XPPEDIT 18 0 "(b-a)" "6#,&%\"bG\"\"\"%\"aG!\"\"" } {TEXT -1 14 ")/20, where " }{XPPEDIT 18 0 "a..b" "6#;%\"aG%\"bG" } {TEXT -1 27 " is the horizontal range (" }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT -1 152 "-range) of the plot. Frequently, it will be necessary to reduce the stepsize, for improved numerical accuracy. This can be \+ done by specifying the option" }}{PARA 256 "" 0 "" {TEXT 297 10 "steps ize=h" }{TEXT 319 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {TEXT 298 1 "h" }{TEXT -1 47 " denotes the desired stepsize, smaller \+ than (" }{XPPEDIT 18 0 "b-a" "6#,&%\"bG\"\"\"%\"aG!\"\"" }{TEXT -1 85 ")/20 . Let's try the following example, where we approximate the \+ same solutions of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " } {XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 47 " as before, but wi th a larger vertical range (" }{XPPEDIT 18 0 "-10..10" "6#;,$\"#5!\"\" F%" }{TEXT -1 58 ") and using the classical Euler method (instead of \+ \"rk4\"):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "DEplot(D(x)(t)=x(t)^2,x(t),t=-3..3,x=-9..9,\{[x(0)=1] ,[x(0)=-1]\},method=classical,linecolor=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6'-%'CURVESG6]dl7%7$$!+YR%e+$!\"*$! +e'fLZ*F*7$$!+ag:%*HF*$!+U.kE&)F*7$$!+&>Lh.$F*$!+F\"Gss)F*7%7$$!+sW0!p #F*F+7$$!+!em$yEF*F07$$!+@PM?FF*F57%7$$!+)*\\EuBF*F+7$$!+1rdiBF*F07$$! +ZUb/CF*F57%7$$!+CbZe?F*F+7$$!+KwyY?F*F07$$!+tZw)3#F*F57%7$$!+]goU%*4\"F*F07$$!+^jRT6F*F57%7$$!+(Gw J&z!#5F+7$$!+ftHOyFapF07$$!+r(ogD)FapF57%7$$!+]:G&z%FapF+7$$!+AESyYFap F07$$!+MS<)4&FapF57%7$$!+8oQP;FapF+7$$!+&)y]?:FapF07$$!+(Hz-%>FapF57%7 $$\"+Cz]?:FapF+7$$\"+_oQP;FapF07$$\"+Sah<7FapF57%7$$\"+hESyYFapF+7$$\" +*e\"G&z%FapF07$$\"+x,^vVFapF57%7$$\"+)R(HOyFapF+7$$\"+Ej<`zFapF07$$\" +9\\SLvFapF57%7$$\"+9#>%*4\"F*F+7$$\"+1r566F*F07$$\"+l*H\"p5F*F57%7$$ \"+)o3_T\"F*F+7$$\"+!e'*oU\"F*F07$$\"+R%>\\Q\"F*F57%7$$\"+i\")*4t\"F*F +7$$\"+agoUFF*F ^y7%7$$!+I\"=dP#F*Fdx7$$!+uR7hBF*Fix7$$!+4_r.CF*F^y7%7$$!+c'G*f?F*Fdx7 $$!++XLX?F*Fix7$$!+Nd#z3#F*F^y7%7$$!+#=RTu\"F*Fdx7$$!+E]aHl\"FapFdx7$$!+tl(f]\"FapFix7$$!+A*))=$>FapF^y7%7$$\"+ 7m(f]\"FapFdx7$$\"+k\"=>l\"FapFix7$$\"+:e+E7FapF^y7%7$$\"+\\8(Qm%FapFd x7$$\"+,H\")4[FapFix7$$\"+_0!RQ%FapF^y7%7$$\"+'3m<#yFapFdx7$$\"+QwqnzF apFix7$$\"+*G&zTvFapF^y7%7$$\"+#3mz4\"F*Fdx7$$\"+Q-c76F*Fix7$$\"+.!p*p 5F*F^y7%7$$\"+cbv89F*Fdx7$$\"+7(\\$G9F*Fix7$$\"+x%edQ\"F*F^y7%7$$\"+I] aHm$4IF*$!+TK6yvF*7 $$!+1Qj!*HF*$!+vITKmF*7$$!+gO4MIF*$!+A&Q,$oF*7%7$$!+?nd$p#F*Fbel7$$!+K V%[n#F*Fgel7$$!+'=/$=FF*F\\fl7%7$$!+YsyxBF*Fbel7$$!+e[0fBF*Fgel7$$!+7Z ^-CF*F\\fl7%7$$!+sx*>1#F*Fbel7$$!+%QlK/#F*Fgel7$$!+Q_s'3#F*F\\fl7%7$$! +)H3iu\"F*Fbel7$$!+5fZF>FapF\\fl7%7$$\"+^aG&[\"Fap Fbel7$$\"+D$4En\"FapFgel7$$\"+\"y5!Q7FapF\\fl7%7$$\"+)=!=VYFapFbel7$$ \"+iS]I[FapFgel7$$\"+=b!fR%FapF\\fl7%7$$\"+D\\2,yFapFbel7$$\"+*z)R))zF apFgel7$$\"+b-!Qb(FapF\\fl7%7$$\"+mp*e4\"F*Fbel7$$\"+a$HY6\"F*Fgel7$$ \"++&p62\"F*F\\fl7%7$$\"+Sko69F*Fbel7$$\"+G)=/V\"F*Fgel7$$\"+u*epQ\"F* F\\fl7%7$$\"+9fZF1#F*Fgel7$$\"+Az`=?F*F\\fl7%7$$\"+i[0fBF*Fbel7$ $\"+]syxBF*Fgel7$$\"+'RFVL#F*F\\fl7%7$$\"+OV%[n#F*Fbel7$$\"+Cnd$p#F*Fg el7$$\"+qo6]EF*F\\fl7%7$$\"+5Qj!*HF*Fbel7$$\"+)>m$4IF*Fgel7$$\"+Wj!f'H F*F\\fl7%7$$!+qGX7IF*$!+ZM5ImF*7$$!+Ira()HF*$!+Fgo&o&F*7$$!+2,HKIF*$!+ W(=3)eF*7%7$$!+'Rjmp#F*F`bm7$$!+cwvrEF*Febm7$$!+L1];FF*Fjbm7%7$$!+AR(3 Q#F*F`bm7$$!+#=ofN#F*Febm7$$!+f6r+CF*Fjbm7%7$$!+[W3l?F*F`bm7$$!+3(y,/# F*Febm7$$!+&o@\\3#F*Fjbm7%7$$!+u\\H\\!)FapF`bm7$$!+=\"3- x(FapFebm7$$!+()yj<#)FapFjbm7%7$$!+\"zq8'[FapF`bm7$$!+\"Q8Bh%FapFebm7$ $!+]Juf]FapFjbm7%7$$!+agZ.FapFjb m7%7$$\"+$o=WX\"FapF`bm7$$\"+$4wMq\"FapFebm7$$\"+Cj/c7FapFjbm7%7$$\"+? 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"iterations=100" }{TEXT -1 225 ", it would take about ten times as long, without any visible improvement! On the other hand, if you tri ed to speed things up by using a stepsize significantly greater than . 001, the plot would not be reliable. Check it out!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Beyond \"rk4\" (the defau lt method) and \"Euler\" ( " }{TEXT 320 16 "method=classical" }{TEXT -1 185 " ), you have a choice of several, much more sophisticated d.e. solvers. One of them is \"rkf45\", a so-called 4th/5th order Runge-Ku tta-Fehlberg method, which can be invoked by specifying" }}{PARA 256 " " 0 "" {TEXT 305 12 "method=rkf45" }}{PARA 0 "" 0 "" {TEXT 306 8 "in t he " }{TEXT 307 6 "DEplot" }{TEXT 308 41 " command. While \"rk4\" an d \"Euler\" use a " }{TEXT 309 5 "fixed" }{TEXT 310 29 " stepsize, \"r kf45\" employs a " }{TEXT 311 8 "variable" }{TEXT 312 332 " stepsize t hat's automatically adjusted in accordance with the \"stiffness\" of t he problem. As an adaptive (self-correcting) method, \"rkf45\" is more reliable than \"rk4\" or \"Euler\", but usually slower. But in \"stif f\" problems (like the previous example), \"rkf45\" offers a double ad vantage: You don't have to worry about the stepsize, " }{TEXT 313 3 "a nd" }{TEXT 314 163 " it is faster than \"rk4\" or \"Euler\"! (In gene ral, controlling and adjusting the stepsize as needed is more efficien t than using a very small stepsize throughout.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 677 "Since \"rkf45\" controls the (numerical) stepsize automatically, specifying a stepsize when us ing this method has no effect on the numerical accuracy of the procedu re; it does, however, affect the number of points plotted and, thereb y, the apparent smoothness of the graph. More precisely, in conjunctio n with \"rkf45\", the specified stepsize becomes simply the horizontal distance between consecutive points plotted. For example, if we solve the previous problem with \"rkf45\", we should specify a stepsize of \+ .1 to obtain the same number of points plotted as before, when we used \"rk4\" with a (numerical) stepsize of .001 and 100 iterations betwee n points plotted. Let's try it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "DEplot(D(x)(t)=1-exp(t)*x(t )^2,x(t),t=0..15,\{[x(0)=0]\},\nmethod=rkf45,stepsize=.1,arrows=none,l inecolor=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6&-%'CURVESG6%7ct7$$\"\"!F)F(7$$\"\"\"!\"\"$\"3'z\"Q<=v@k**!#>7$$\" \"#F-$\"3[&*G&>Wc&p>!#=7$$\"\"$F-$\"3ks!y)zS#>*GF67$$\"\"%F-$\"39/I=C[ 'Qt$F67$$\"\"&F-$\"3u`ayTlDnWF67$$\"\"'F-$\"3aeL\\twpp]F67$$\"\"(F-$\" 3XJ69<@-GbF67$$\"\")F-$\"3([0]p>!3SeF67$$\"\"*F-$\"3JvK0ZfH9gF67$$\"#5 F-$\"3'e<&)4;2t1'F67$$\"#6F-$\"3%y'Go@0e?gF67$$\"#7F-$\"3y+@_\"y9q*eF6 7$$\"#8F-$\"3'eLj6ei#=dF67$$\"#9F-$\"36[uAum1.bF67$$\"#:F-$\"3y;-kgvam _F67$$\"#;F-$\"3d&)HE+B>?]F67$$\"#F-$\"3MySf`A3#H%F67$$\"#?F-$\"3')R>h'\\\\=Uf4$F67$$\"#EF-$\"3K2\"H$45.LH F67$$\"#FF-$\"3O#)3SJSFzFF67$$\"#GF-$\"3p!)3F&p@Tj#F67$$\"#HF-$\"3'pBP @rZq\\#F67$$\"#IF-$\"3]9AW_cbnBF67$$\"#JF-$\"3mY9-L5=XAF67$$\"#KF-$\"3 #)Q.An4\\H@F67$$\"#LF-$\"3'Rq'e?E3??F67$$\"#MF-$\"3BzQ-y6e;>F67$$\"#NF -$\"3.\">ttpP'==F67$$\"#OF-$\"32p3cJq#fs\"F67$$\"#PF-$\"3#y'=#e*e9Q;F6 7$$\"#QF-$\"3K%))yE56]b\"F67$$\"#RF-$\"3y'e\\Q)zDw9F67$$\"#SF-$\"3[\"G V\"o)Q;S\"F67$$\"#TF-$\"3Wf`Lp=#4L\"F67$$\"#UF-$\"3!H#eo\\'*)QE\"F67$$ \"#VF-$\"3;k.#)\\$Q.?\"F67$$\"#WF-$\"3u[s&)4m2S6F67$$\"#XF-$\"3s%)>9wY #H3\"F67$$\"#YF-$\"3#>!e%\\e8(G5F67$$\"#ZF-$\"3q*\\E?:WGx*F07$$\"#[F-$ \"3'\\[AD.w[G*F07$$\"#\\F-$\"3!f$G(Q8B=#))F07$$\"#]F-$\"3wtUL(eeBQ)F07 $$\"#^F-$\"3)ec[inJ_'zF07$$\"#_F-$\"3!*Gi(=\"HEpvF07$$\"#`F-$\"3WKo!Gs RL>(F07$$\"#aF-$\"3s#>hBR6k$oF07$$\"#bF-$\"3%Q;!G'H'[(\\'F07$$\"#cF-$ \"3/_YX:wivhF07$$\"#dF-$\"3%Q`D\"=,&*peF07$$\"#eF-$\"3+9i=SqhzbF07$$\" #fF-$\"3JNA1zr$QI&F07$$\"#gF-$\"3eW/e'=i=/&F07$$\"#hF-$\"39>f$[/%)Hz%F 07$$\"#iF-$\"3]9!Q0rKlb%F07$$\"#jF-$\"3:]3n9R(=L%F07$$\"#kF-$\"35D:[Ru S=TF07$$\"#lF-$\"3W0qq7Sc:RF07$$\"#mF-$\"3C/.)*)30Gs$F07$$\"#nF-$\"3E# \\95@?'RNF07$$\"#oF-$\"3u!RQVkDbO$F07$$\"#pF-$\"3+,(z0*G1+KF07$$\"#qF- $\"3P#pqHH(zUIF07$$\"#rF-$\"3E1CPEnJ$*GF07$$\"#sF-$\"3OE$)G!oI7v#F07$$ \"#tF-$\"3)\\p'=C!ohh#F07$$\"#uF-$\"3mC-PAtx([#F07$$\"#vF-$\"3i$)*=K) \\slBF07$$\"#wF-$\"3k*)p&3[%p\\AF07$$\"#xF-$\"3Mi,ETaQR@F07$$\"#yF-$\" 32>9^$y7X.#F07$$\"#zF-$\"3<\"*fMai![$>F07$$\"#!)F-$\"3>g6%)4&3+%=F07$$ \"#\")F-$\"3#Q@EL(e()\\y?@1#[h8F07$$\"#()F-$\"3CV&eW6m[H\"F07$$\"#))F-$\"3RQEr^'>:B \"F07$$\"#*)F-$\"3G\\VVE7Gr6F07$$\"#!*F-$\"3--JQox*R6\"F07$$\"#\"*F-$ \"3?zfTaJ_f5F07$$\"##*F-$\"3)GvfE1>x+\"F07$$\"#$*F-$\"3WjP&f8RXe*!#?7$ $\"#%*F-$\"39#pn#)oEg6*Fhhl7$$\"#&*F-$\"3+u1ME(o/n)Fhhl7$$\"#'*F-$\"3; _Q8k7tY#)Fhhl7$$\"#(*F-$\"3Y33E0GuVyFhhl7$$\"#)*F-$\"3s%)opOK[guFhhl7$ $\"#**F-$\"3#p+m#=I)f4(Fhhl7$$\"$+\"F-$\"3m2[&eTD$\\nFhhl7$$\"$,\"F-$ \"3Ut#HvbF'>kFhhl7$$\"$-\"F-$\"3__^1(Gfg5'Fhhl7$$\"$.\"F-$\"3z#=`Q0Hy! eFhhl7$$\"$/\"F-$\"3k\\Diw!)=CbFhhl7$$\"$0\"F-$\"3%)3*>,!eTa_Fhhl7$$\" $1\"F-$\"31FSfiN$y*\\Fhhl7$$\"$2\"F-$\"3UoL7z_z`ZFhhl7$$\"$3\"F-$\"3\\ 5kd#4Qh?NFhhl7$$\"$9\"F-$\"3uRtA]ww[LFhhl7$$\"$:\"F-$\"3E,xPa [J&=$Fhhl7$$\"$;\"F-$\"3?i;%3v[)HIFhhl7$$\"$<\"F-$\"3%=Hk5^u>)GFhhl7$$ \"$=\"F-$\"3NI\\]YEKTFFhhl7$$\"$>\"F-$\"3Yvb\"ppRvg#Fhhl7$$\"$?\"F-$\" 31ZmR)e*G![#Fhhl7$$\"$@\"F-$\"3a;TkLIDfBFhhl7$$\"$A\"F-$\"3g^Eg9j7WAFh hl7$$\"$B\"F-$\"3()=9[m/iM@Fhhl7$$\"$C\"F-$\"3WI@:O;YI?Fhhl7$$\"$D\"F- $\"3oGA)4U'QJ>Fhhl7$$\"$E\"F-$\"3*eb]$*=[r$=Fhhl7$$\"$F\"F-$\"3;HO%yQ6 vu\"Fhhl7$$\"$G\"F-$\"31x\"e7E[Am\"Fhhl7$$\"$H\"F-$\"3m1(*>Eq9\"e\"Fhh l7$$\"$I\"F-$\"3&)[j\"*HR+/:Fhhl7$$\"$J\"F-$\"35a'QdYF1V\"Fhhl7$$\"$K \"F-$\"3Mp^>02$3O\"Fhhl7$$\"$L\"F-$\"3CE6%4%4W%H\"Fhhl7$$\"$M\"F-$\"32 p_F:3HJ7Fhhl7$$\"$N\"F-$\"3utZpeBAr6Fhhl7$$\"$O\"F-$\"3d5EsKd396Fhhl7$ $\"$P\"F-$\"3#pb\">7mtf5Fhhl7$$\"$Q\"F-$\"3.R&y'*RR!35Fhhl7$$\"$R\"F-$ \"3_jcM](\\')e*!#@7$$\"$S\"F-$\"3EE%GFK**37*F_gm7$$\"$T\"F-$\"3evW!\\. rfn)F_gm7$$\"$U\"F-$\"3Ktfv&y^FD)F_gm7$$\"$V\"F-$\"3g\"4[8q!=]yF_gm7$$ \"$W\"F-$\"3uzuuNaCnuF_gm7$$\"$X\"F-$\"3*z`Pn50I5(F_gm7$$\"$Y\"F-$\"3% )4@pZ*Glv'F_gm7$$\"$Z\"F-$\"330'\\![h&pU'F_gm7$$\"$[\"F-$\"3X6q'p+iM6' F_gm7$$\"$\\\"F-$\"3%)pVa-5E:eF_gm7$$\"$]\"F-$\"3#>Qd&)p;;`&F_gm-%&STY LEG6#%%LINEG-%*THICKNESSG6#F9-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;$!++++ +v!#5$\"++++v:!\");$!+!e`O.$!#6$\"+>DnqjFh[n-%+AXESLABELSG6$Q\"t6\"Q%x (t)Ff\\n" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 6 "DEplot" }{TEXT -1 22 " will not only solve " }{TEXT 316 11 "first-or der" }{TEXT -1 11 ", but also " }{TEXT 317 12 "higher-order" }{TEXT -1 157 " i.v.p.s (although no direction fields can be generated for th e latter). The following command, for example, produces an approximate solution of the i.v.p. " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 "' ' + " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "x(0)=2" "6#/-%\"xG6#\"\"!\"\"#" }{TEXT -1 44 ", x'(0) \+ = -3, whose symbolic solution is " }{XPPEDIT 18 0 "2*cos(t)-3*sin(t) " "6#,&*&\"\"#\"\"\"-%$cosG6#%\"tGF&F&*&\"\"$F&-%$sinG6#F*F&!\"\"" } {TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot(D(D(x))(t)+x(t)=0,x(t),t=-10..20,\{[x(0)= 2,D(x)(0)=-3]\},stepsize=.2,linecolor=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6&-%'CURVESG6%7ct7$$!$+\"!\"\"$!3$= jX2R1-J$!#<7$$!#)*F*$!3)*Q$f=$**GgHF-7$$!#'*F*$!3K]h7bgN#\\#F-7$$!#%*F *$!3A*[gv!)f]#>F-7$$!##*F*$!3%\\rX.]<5G\"F-7$$!#!*F*$!3A$*=xy10fe!#=7$ $!#))F*$\"3\\q*pr]bcK\"FG7$$!#')F*$\"39>&*Q\"*>^d%)FG7$$!#%)F*$\"3rTQ< ;%>__\"F-7$$!##)F*$\"3aNZ>[>)Q9#F-7$$!#!)F*$\"3KGFNsY2xEF-7$$!#yF*$\"3 )*yGKx3a.JF-7$$!#wF*$\"3>K]I,(yiS$F-7$$!#uF*$\"3Y*e,E%*=Kd$F-7$$!#sF*$ \"3w?Dg?iq(f$F-7$$!#qF*$\"3Y8!3[Ik(yMF-7$$!#oF*$\"3UCy0M]8@KF-7$$!#mF* $\"3(pZ!Qu#*3NGF-7$$!#kF*$\"3%zIw>XZpMF-7$ $!#UF*$!3/xx#*f*[_f$F-7$$!#SF*$!3i39kFZpxNF-7$$!#QF*$!3dLSj'44vT$F-7$$ !#OF*$!3sd)=D;y57$F-7$$!#MF*$!3VEq>\"p>-q#F-7$$!#KF*$!3%*eU#=)>rr@F-7$ $!#IF*$!3G30(*o\\ic:F-7$$!#GF*$!3Cko=3B![z)FG7$$!#EF*$!3-+%[A\"Rts;FG7 $$!#CF*$\"3o&o`h5@g^&FG7$$!#AF*$\"3)>p')p(p[[7F-7$$!#?F*$\"33iXT2')f&* =F-7$$FGF*$\"3i?HF.(QrY#F-7$$!#;F*$\"3-*>RXw@.%HF-7$$!#9F*$\"37-kxvMG' H$F-7$$!#7F*$\"3I'yfowK3_$F-7$$!#5F*$\"3=7-;mv,0OF-7$$!\")F*$\"3s0,R\" p\"[XNF-7$$!\"'F*$\"315-+]')fWLF-7$$!\"%F*$\"3ggv#\\,x.,$F-7$$!\"#F*$ \"3q1]1[69cDF-7$$\"\"!F_[l$\"\"#F_[l7$$Fa[lF*$\"3?LLLLL8k8F-7$$\"\"%F* $\"3>lmmm)\\)QnFG7$$\"\"'F*$!3=\"G'\\j,xAV!#>7$$\"\")F*$!3kD\"4b\\khe( FG7$$\"#5F*$!3?msdESZ9c6O ;EF-7$$\"#;F*$!3w2M8R)oq0$F-7$$\"#=F*$!3%*y8w\"p.fP$F-7$$\"#?F*$!35\"o =5Ab,c$F-7$$\"#AF*$!3b.tf9#yCg$F-7$$\"#CF*$!3p`,yeb=,NF-7$$\"#EF*$!3Q6 [a\"\\:.E$F-7$$\"#GF*$!3_**R)*41Z*)GF-7$$\"#IF*$!3!)[(R.0NMS#F-7$$\"#K F*$!3#)[[E-^e@=F-7$$\"#MF*$!3!*ox\"Qv;r;\"F-7$$\"#OF*$!3Yo(f(='47m%FG7 $$\"#QF*$\"3b3MHo[cMDFG7$$\"#SF*$\"3TYj,!e!HH'*FG7$$\"#UF*$\"3w6Jc\"38 Sj\"F-7$$\"#WF*$\"3@&>I>Ib*RAF-7$$\"#YF*$\"3U&y?hT*fcFF-7$$\"#[F*$\"3# \\Qr83\\L;$F-7$$\"#]F*$\"3Y%o.8+*)RW$F-7$$\"#_F*$\"3ckQ_d8L(e$F-7$$\"# aF*$\"3a)p2j\"=m(e$F-7$$\"#cF*$\"3(eNV,Ln\\W$F-7$$\"#eF*$\"3C$>QGlO\\; $F-7$$\"#gF*$\"37Ja&yaL(eFF-7$$\"#iF*$\"3D0)38t^DC#F-7$$\"#kF*$\"3V&*Q N#Hppj\"F-7$$\"#mF*$\"3Yic*oF$Gh'*FG7$$\"#oF*$\"3%=2k$[\\snDFG7$$\"#qF *$!3X8t!*p9>GYFG7$$\"#sF*$!3)H$o00&fR;\"F-7$$\"#uF*$!3we\"fo3(p==F-7$$ \"#wF*$!3t[&*e2.$4S#F-7$$\"#yF*$!3O0Rf[![u)GF-7$$\"#!)F*$!3sBr@%)\\&)e KF-7$$\"##)F*$!3=PpD)yW.]$F-7$$\"#%)F*$!3*\\4\"o&R!H-OF-7$$\"#')F*$!3] +Y_)yF1c$F-7$$\"#))F*$!3k]YA1!=qP$F-7$$\"#!*F*$!3I!o?>#4yeIF-7$$\"##*F *$!3y`4)\\U.'=EF-7$$\"#%*F*$!3G1r[VO.u?F-7$$\"#'*F*$!3ck@3q8yY9F-7$$\" #)*F*$!3q\"RTU!e_=wFG7$$\"$+\"F*$!3QFe4\\ZDbYF`\\l7$$\"$-\"F*$\"3$eY#> lq-1nFG7$$\"$/\"F*$\"3!4v*yCK-h8F-7$$\"$1\"F*$\"3++G%p([=(*>F-7$$\"$3 \"F*$\"3K*\\%3Hms`DF-7$$\"$5\"F*$\"3UfdMz:Y3IF-7$$\"$7\"F*$\"34*4tKXhK M$F-7$$\"$9\"F*$\"3Oj'*4 A%HF-7$$\"$C\"F*$\"3%enN'*ed&pCF-7$$\"$E\"F*$\"3U!eKfya%)*=F-7$$\"$G\" F*$\"3SbMG-\"p;D\"F-7$$\"$I\"F*$\"3=kUMGA&)\\bFG7$$\"$K\"F*$!3pV[m/'H# Q;FG7$$\"$M\"F*$!3')Gacvd*4w)FG7$$\"$O\"F*$!3)ek8(p\"\\Mb\"F-7$$\"$Q\" F*$!3#pd*\\'po)o@F-7$$\"$S\"F*$!3mry&=.Cyp#F-7$$\"$U\"F*$!3&ejcS.G#>JF -7$$\"$W\"F*$!3LB)R!)>\"G;MF-7$$\"$Y\"F*$!3Ue]/x89xNF-7$$\"$[\"F*$!3oB ImueR&f$F-7$$\"$]\"F*$!3w2R>(3<.Z$F-7$$\"$_\"F*$!3^-*4^]\"*o?$F-7$$\"$ a\"F*$!3Bz\"ew\"4i:GF-7$$\"$c\"F*$!37k)Rfu.@J#F-7$$\"$e\"F*$!3%QoV^>8k r\"F-7$$\"$g\"F*$!3WiI\">tKGF-7$$\"$u\"F*$\"3I 8l\"3Mq$>KF-7$$\"$w\"F*$\"33\\Kv4dmxMF-7$$\"$y\"F*$\"3oh`T')4K(f$F-7$$ \"$!=F*$\"3)f;@A9mNd$F-7$$\"$#=F*$\"3x:VX4$[tS$F-7$$\"$%=F*$\"3ZUz.=SH 0JF-7$$\"$'=F*$\"3WHFr=]WzEF-7$$\"$)=F*$\"3)p)[VC#yn9#F-7$$\"$!>F*$\"3 )=]'43*G&G:F-7$$\"$#>F*$\"3uTu%*z8W$\\)FG7$$\"$%>F*$\"33P;d,?+j8FG7$$ \"$'>F*$!3hJ**G6#o<#eFG7$$\"$)>F*$!3cqhfsUWx7F-7$$\"$+#F*$!3npb'Rz%y@> F--%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-%'COLOURG6&%$RGBGF_[lF_[lF_[l -%%VIEWG6$;$!++++]6F[z$\"++++]@F[z;$!+/I&G'R!\"*$\"+bBRlRF]\\n-%+AXESL ABELSG6$Q\"t6\"Q%x(t)Fd\\n" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "Note that the two initial conditions appear inside a " }{TEXT 322 23 "single pair of brackets" }{TEXT -1 25 " (because t hey determine " }{TEXT 323 23 "only one solution curve" }{TEXT -1 240 "). The stepsize specification was necessary, due to the oscillatory n ature of the graph and the length of the plot interval; the default st epsize would have been 30/20 = 1.5 ! (Check out what you would get w ithout stepsize specification.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 278 "Basically, you have the same options as \+ in the first-order case (except for those relating to direction fields , of course), and just as in the first-order case, several approximate solutions can be generated simultaneously, by specifying several sets of initial data in the form" }}{PARA 256 "" 0 "" {TEXT 321 74 "\{[x(t 0)=x0,D(x)(t0)=v0],[x(t1)=x1,D(x)(t1)=v1],[x(t3)=x3,D(x)(t3)=v3],...\} " }{TEXT 272 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 205 "Note that, in anal ogy to the first-order case, the entire set of initial conditions is e nclosed in curly braces, while the initial conditions for each individ ual solution curve appear in brackets. Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "DEplot(D(D(x ))(t)+x(t)=0,x(t),t=-10..20,\{[x(0)=2,D(x)(0)=-3],[x(0)=-3,D(x)(0)=2] \},linecolor=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 {PLOTDATA 2 "6&-%'CURVESG6&777$$!+++++5!\")$\"3_4\\+4oD0O!#<7$$!+++++& )!\"*$\"3=? jr'G\\rF47$$!+++++SF1$\"3e'REK&e`uMF-7$$!+++++DF1$\"3o*o)Qel[17F-7$$F) F1$!3NiwA())[QI$F-7$$\")+++]F*$!3ummmm\"zWn\"F-7$$\"+++++?F1$\"3E\"e9 \"4NE\\GF-7$$\"+++++NF1$\"3S`P^CQzt>F-7$$\"+++++]F1$!3/g+$pAKg=#F-7$$ \"+++++lF1$!33nF8')43D@F-7$$\"+++++!)F1$\"3W,=]&\\8Ad\"F-7$$\"+++++&*F 1$\"3%)e:m#yK-\"F-7$$\"++++]7F*$!3yPh%fI^ Y3#F-7$$\"+++++9F*$\"3y')**o!\\Li[&F47$$\"++++]:F*$\"3Uk$)=zL!>%>F-7$$ \"+++++\"F-7$F6$\"3o:!3[Ik(yMF -7$F;$!3+tbH#G9G*pF47$F@$!3%*49kFZpxNF-7$FE$\"3\">aZE,#HJ>F47$FJ$\"328 -;mv,0OF-7$FN$\"3![LLLL$3xJF47$FS$!3_mT&)ehRRLF-7$FX$!37)3mN64ab)F47$F gn$\"3f'>I'\\Bj7GF-7$F\\o$\"3-jA2E)z;C\"F-7$Fao$!39YH)\\V)RzAF-7$Ffo$! 3;JBV')=E#\\\"F-7$F[p$\"3yI?@b*GPw\"F-7$F`p$\"35c^sX*>di\"F-7$Fep$!3!y NEDPdPG\"F-7$Fjp$!3_.,.uv]h;F-7$F_q$\"3Tupg!Q;@_)F47$Fdq$\"3hZ/$=#R0>; F-7$Fiq$!3UhhG:/LqZF4-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-%'COLOURG6 &%$RGBG\"\"!FhvFhv-%%VIEWG6$;$!++++]6F*$\"++++]@F*;$!+0B%o$RF1$\"+'Q/W 'RF1-%+AXESLABELSG6$Q\"t6\"Q%x(t)Fjw" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "Pretty lousy! Obviously, we should redu ce the stepsize or use a more flexible solver. 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