{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 3 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 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 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 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 68 "Slope fields and the Eul er method for solving differential equations" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The Euler method uses the slope fields to approximate solutions of differential equations." }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Slope fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "For differential equations of the form" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(y(x),x)" "6#-%%d iffG6$-%\"yG6#%\"xGF)" }{TEXT -1 27 " = expression in x and y" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "a slope field can be computed in a given regio n (usually a rectangle) by computing values, at a big number of poin ts filling the region in question, of the expression on the right hand side of the above equation. Maple does that automatically with the \+ package " }{MPLTEXT 1 0 7 "DEtools" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Consider the following ex amples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "DEplot(diff(y(x),x)=x+y(x)+1,y(x),x=-10..10,[[y(0)=-4]],y=-10..10,ste psize=.01,dirgrid=[10,10],color=red,linecolor=sin(x*Pi));" }}{PARA 13 "" 1 "" {GLPLOT2D 380 380 380 {PLOTDATA 2 "6&-%'CURVESG6aq7%7$$!+q)Re+ \"!\")$!+MYU!*))!\"*7$$!+(H,;%**F-$!+Pv&46\"F*7$$!+O[\\*o*F-$!+Z:Xh5F* 7%7$$!+Cc)Q%yF-$!+Ws&3*))F-7$$!+K*p;r(F-$!+wU\"46\"F*7$$!+!y4IY(F-$!+! 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F^jp$!3'4+X&G#Hw>\"Fa[q7$$\"##)F^jp$!3+*)QPkBXn6Fa[q7$$\"#%)F^jp$!3KX: i!4&yQ6Fa[q7$$\"#')F^jp$!3qN@8M[^66Fa[q7$$\"#))F^jp$!3a`!eur.\"Fa[q7$$\"#%* F^jp$!3;%H%>y`^95Fa[q7$$\"#'*F^jp$!3G!z>e:y)G**Fajp7$$\"#)*F^jp$!3jZI0 nFu@(*Fajp7$$\"$+\"F^jp$!3#fy%37P " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Euler's method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Euler's method to find solutions to differential equation s is the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 45 "1. Form a slope field with a given step size." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "2. Start from \+ a point corresponding to the given initial condition. Continue a shor t distance in the direction given by the slope field at that point. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "3. Re calculate the slopefield, and repeat step 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "In this way Maple's dsolve c ommand (with the choices numeric and method=classical[foreuler]) so lves differential equations. (That command has also much more powerfu l methods.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "The solution graph provided by the DEplot command (as in the previous section) uses better methods than the classical Euler me thod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " Here are some examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "p lot0:=odeplot(dsolve(\{diff(y(x),x)=cos(x),y(0)=0\},\{y(x)\},numeric,m ethod=classical[foreuler],stepsize=1),[x,y(x)],-10..10,color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot1:=DEplot(diff(y(x),x)=cos(x ),y(x),x=-10..10,[[y(0)=0]],y=-2..2,arrows=SMALL,dirgrid=[10,10],linec olor=black,stepsize=1):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the n ame changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot2:=plot(sin(x),x=-10..10,y=-2..2,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(\{plot0,plot2\},thic kness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 449 449 449 {PLOTDATA 2 "6'-%' CURVESG6$7T7$$!#5\"\"!$\"3-,l)o)HA%R&!#=7$$!+Nn$=f*!\"*$\"3h:ckQz^?;F- 7$$!+qMn$=*F1$!3#)y?wUHUICF-7$$!+0-^v()F1$!3S2AfN:/$4'F-7$$!+SpMn$)F1$ !3O;X2r*zbw)F-7$$!+vO=fzF1$!32?U\"[x%*3+\"!#<7$$!+5/-^vF1$!3yHD.gI')=' *F-7$$!+Xr&G9(F1$!37zX(pr:%fwF-7$$!+!)QpMnF1$!3a/vvrt`_WF-7$$!+:1`EjF1 $!3cKz:.14^_!#>7$$!+]tO=fF1$\"3y3LGhQgxMF-7$$!+&3/-^&F1$\"3'[+'f))p'z* oF-7$$!+?3/-^F1$\"3e\"H*>SV-u\"*F-7$$!+bv(Qp%F1$\"38Nt(\\8E=$**F-7$$!+ !H9dG%F1$\"3Woqz#ono/*F-7$$!+D5bxQF1$\"35#*3s_^akmF-7$$!+gxQpMF1$\"3!H YLRRni<$F-7$$!+&\\C71$F1$!3#**\\XSI9&[%)Fgn7$$!+I71`EF1$!3X!=A\"yG:QZF -7$$!+lz*[C#F1$!3-x`FS&yR'yF-7$$!++ZtO=F1$!3\">xft5k(3(*F-7$$!+N9dG9F1 $!3mW,'=`;%p**F-7$$!+q\"3/-\"F1$!3Nve8e86.')F-7$$!+Z!\\C7'F)$!3id.$olG V$eF-7$$!+%R;3/#F)$!3mje9)G!>F?F-7$$\"+fi\"3/#F)$\"33+++fi\"3/#F-7$$\" +7*[C7'F)$\"3mV(**=#\\uPgF-7$$\"+c\"3/-\"F1$\"3A')H\"Q_*)zP*F-7$$\"+@9 dG9F1$\"3mnsrLdF^6FH7$$\"+'oMn$=F1$\"3tX4+07847FH7$$\"+^z*[C#F1$\"3=zV /w)f=5\"FH7$$\"+;71`EF1$\"3o9A$fAh3Z)F-7$$\"+\"[C71$F1$\"37vls7bom[F-7 $$\"+YxQpMF1$\"3W-E\"))*[F#)zFgn7$$\"+65bxQF1$!3#4d)4/n2mIF-7$$\"+wUr& G%F1$!3;W#*)p#GL\"4'F-7$$\"+Tv(Qp%F1$!3G')G&*f)*\\!y(F-7$$\"+13/-^F1$! 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