{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 255 0 0 1 2 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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 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 "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "New Page" -1 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 3 258 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 257 13 "template2.mws" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 479 "This worksheet is designed to give you a template for many of \+ the things you will be doing with systems of differential equations. D elete uneccessary lines, and don't forget to execute the worksheet fro m the top once you have entered all the desired changes. You may find it useful to keep a completed Maple worksheet open in another window \+ as a guide (\"tile\" the windows so you can see both at once). In fac t you can even copy and paste from one window to another if you wish. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 257 " " 0 "" {TEXT -1 15 "Getting started" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Every Maple worksheet should begin by re-initializing the Mapl e \"kernel\" and loading the additional packages that we are most like ly to use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 256 18 "Defining the model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "All dif ferential equation models begin with a differential equation. If the p roblem is an intial value problem, an initial condition is also needed . Replace the question marks ( " }{TEXT 0 1 "?" }{TEXT -1 67 " ) in th e following input regions to define the relevant ODE model." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ode1 := diff( %?, %? ) = %? \+ ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ode2 := diff( %?, %? ) = %? ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "MODEL := \{ ode1 , ode2 \} ; # define the differential equation model" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "VAR := \{ %?, %? \} ; \+ # identify the variables in the model" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT 267 10 "Nul lclines" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "It is difficult to giv e a general outline for the determination of nullclines. The key is to set the RHS of each ODE in the system equal to zero and to find all c urves on which the resulting equation is satisfied." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 268 12 "x-nullclines" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 69 "-nullclines are found by setting the RHS of the first ODE equal to 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor( rhs(ode1) ) = 0 ;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 90 "-nullcline(s) typically consists of curves that can be e xpressed either as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 23 " or as a function of " }{XPPEDIT 18 0 "y" "6#%\"yG" } {TEXT -1 4 " : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "xNULL1 \+ := plot( %?, x = %? .. %?, color=GREEN ): # for a nullcline o f the form y=F(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "xNULL2 := plo t( [ %?, y, y = %? .. %? ], color=GREEN ): # for a nullcline of the \+ form x=G(y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xNULLplot := displa y( [ xNULL1, xNULL2 ], axes=BOXED ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xNULLplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 269 12 "y-nullclines" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 69 "-n ullclines are found by setting the RHS of the first ODE equal to 0:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor( rhs(ode2) ) = 0 ; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "y" "6#% \"yG" }{TEXT -1 90 "-nullcline(s) typically consists of curves that ca n be expressed either as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 23 " or as a function of " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 4 " : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "yNULL1 := plot( ?, x = ? .. ?, color=BLUE ): # for a nullcline of \+ the form y=F(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "yNULL2 := plot( [ ?, y, y = ? .. ? ], color=BLUE ): # for a nullcline of the form \+ x=G(y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "yNULLplot := display( [ \+ yNULL1, yNULL2 ], axes=BOXED ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " yNULLplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The complete nullcline plot is the union \+ of the plots of the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 "- and \+ " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 12 "-nullclines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "NULLplot:=display( [xNULLplot, yNUL Lplot]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " title =\"Nullclines for %?\" ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "NULLplo t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 21 "Equilibrium Solutions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "Equilibrium solutions, which occur wherever the nullclin es for the different equations intersect, can sometimes be found by di rectly setting the RHS's to zero and solving the resulting system of e quations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "EQUILeqns := \+ \{ rhs( ode1 ) = 0, rhs( ode2 ) = 0 \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "EQUILsoln := solve( EQUILeqns, VAR ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 54 "Direction Field, Phase Portrait, and Solution Curves (" } {HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "Replace the question marks ( " }{TEXT 0 1 "?" } {TEXT -1 80 " ) in the following input regions to create the directio n field for your model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "IC := [ %? = %?, %? = %? ] ; # specify an initial cond ition (not needed for direction field)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "DOMAIN := %? = %? .. %? ; # specify reasonable interval for indep var" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "RANGE := %? = %? .. %?, %? = %? .. %? ; # specify reasonable intervals for all dep vars" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 15 "Direction F ield" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plotDIR := DEplot( M ODEL, VAR, DOMAIN, RANGE, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " title = \"Direction Field for %?\" ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plotDIR;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 263 14 "P hase Portrait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plotPHASE : = DEplot( MODEL, VAR, DOMAIN, [ IC ], RANGE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " scene=[ %? , %? ], arrows=NONE, \+ linecolor=BLUE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " \+ title = \"Phase Portrait for %?\" ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plotPHASE;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 262 15 "Solution Curves" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P1:= DEplot( MODEL, VAR, DOM AIN, [ IC ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " scene= [ %?, %? ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P2:= DEplot( MODEL, VAR, DOMAIN, [ IC ]," }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 62 " scene=[ %?, %? ], arrows=NONE, \+ linecolor=GREEN ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "displ ay( [ P1, P2 ] , title = \"%? and %? vs. %?\" ); # combined solution c urves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display( array([P1 ,P2]), title = \"%?\");\010 # side-by-side solution curves" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 20 "Analytic Solutions (" }{HYPERLNK 17 "dsolve" 2 "dsolv e" "" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 385 "Occasiona lly it is possible to get an analytic (formula) solution. There's no \+ harm in asking, but you may end up staring at nothing for a long time, or Maple may come back with no answer, or an answer that is completel y unintelligible. Here too, when we learn specific methods, we will \+ be able to ask Maple to use precisely those methods. The command is c omplete by replacing the " }{TEXT 0 1 "?" }{TEXT -1 95 " by dependen t variable ( independent variable), as in the sugar example, somethin g like S(t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "MODEL;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "IC;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "VAR;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IVP := MODEL union convert( IC, set );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "SOLN := dsolve( IVP, VAR );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 21 "Num erical Solutions (" }{HYPERLNK 17 "dsolve,numeric" 2 "dsolve,numeric" "" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "Euler's meth od is not, in general, needed. Maple's approximate solutions (as gener ated by DEplot) are sufficient for our needs. But, if you really want \+ to use Euler's method, here it is." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 1 " " }{TEXT 0 6 "Euler2" }{TEXT 266 42 " (a custom-defined procedure for our uses)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "DO NOT CHANGE ANYTHING IN THE FO LLOWING INPUT REGION - YOU HAVE BEEN WARNED !!!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Euler2 := proc( MODEL, IC, DOMAIN, N )" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 73 " local a, b, dt, dx, dy, f, g, i, ode1, ode2, tt, xx, yy, LISTpts, vars;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " od e1 := op(1,MODEL);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " ode2 := op( 2,MODEL);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " vars := (op(2,lhs(od e1) ), op([1,0],lhs(ode1) ), op([1,0],lhs(ode2) ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " f := unapply( rhs(ode1), vars );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " g := unapply( rhs(ode2), vars );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " a := op(1,rhs(DOMAIN)); b := op(2,rhs(D OMAIN));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " dt := (b-a)/N;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " tt := a;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " xx := eval( vars[2](a), IC );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " yy := eval( vars[3](a), IC );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " #yy := rhs(IC);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " LISTpts := [ tt, xx, yy ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " for i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " dx : = evalf( f(tt,xx,yy) ) * dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " \+ dy := evalf( g(tt,xx,yy) ) * dt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " xx := xx + dx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " yy := yy + dy;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " tt := tt + dt;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " LISTpts := LISTpts, [ tt, xx, y y ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " RETURN( [ LISTpts ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 79 "DO NOT CHANGE ANY THING IN THE PRECEDING INPUT REGION - YOU HAVE BEEN WARNED !!!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Let's recall what we are working with." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "MODEL;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "IC;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "VAR; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "DOMAIN ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The number of steps to be used in Euler's method is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Nsteps := %?; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The points generated by Euler's method are" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ptsE := Euler2( MODEL, IC, D OMAIN, Nsteps );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "These data points can be used to c reate an approximate phase portrait as follows:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "ptsXY := [ seq( [ pt[2], pt[3] ], pt=ptsE ) ]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plotXY := plot( ptsXY, \+ color = GREEN, thickness = 2 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p lotXY;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Another way to view the approximate solution i s to overlay the computed solution on the direction field (and Maple's phase portrait)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "displa y( [ plotDIR, plotPHASE, plotXY ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "In a similar \+ way, the two solution curves can be created:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ptsTX := [ seq( [ pt[1], pt[2] ], pt=ptsE ) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plotTX := plot( ptsTX, co lor = GREEN, thickness = 2 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plo tTX;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ptsTY := [ seq( [ pt[1], pt[3] ], pt=ptsE ) ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plotTY := plot( ptsTY, color = GREEN, thickness = 2 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plotTY ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "And, again together with Maple's solution curve" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display( [ plotTX, P1 ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display( [ plotTY, P2 ] ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}}{MARK "0 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }