{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Couri er" 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 "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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bul let Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 7 "Bug fix" }{TEXT -1 13 ": D Eplot bugs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "There are a number of bugs and weaknesses affecting " } {MPLTEXT 0 21 6 "DEplot" }{TEXT -1 8 " in the " }{MPLTEXT 0 21 7 "DEto ols" }{TEXT -1 9 " package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{PARA 15 "" 0 "" {TEXT -1 67 "It uses thick cur ves (thickness 3) to plot trajectories, even when " }{MPLTEXT 0 21 9 " thickness" }{TEXT -1 27 " is specified as 0, 1 or 2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "DEplot(\{diff(x(t),t)=1,diff(y(t),t)=x(t )\},[x(t),y(t)],t=0..2,\{[x(0)=0,y(0)=1]\},linecolour=blue,thickness=1 ,\narrows=none);" }}}{PARA 0 "" 0 "" {TEXT -1 25 " A work-around is to use " }{MPLTEXT 0 21 4 "subs" }{TEXT -1 18 " on the result of " } {MPLTEXT 0 21 6 "DEplot" }{TEXT -1 12 " as follows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(THICKNESS(3)=THICKNESS(1),\");" }}} {PARA 0 "" 0 "" {TEXT -1 41 "This bug has been corrected in Release 5. " }}{PARA 15 "" 0 "" {TEXT -1 77 "It produces an error message in plot s of 2 by 2 autonomous systems where the " }{MPLTEXT 0 21 5 "scene" } {TEXT -1 47 " option involves the independent variable, and " } {MPLTEXT 0 21 11 "arrows=none" }{TEXT -1 19 " is not specified. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "DEplot(\{diff(x(t),t)=1, dif f(y(t),t)=x(t)\}, [x(t),y(t)], t=0..1, [[x(0)=1,y(0)=1]], scene=[t,x]) ;" }}{PARA 8 "" 1 "" {TEXT -1 61 "Error, (in DEtools/DEplot/CheckDE) i nvalid subscript selector" }}}{PARA 0 "" 0 "" {TEXT -1 46 " A directio n field only makes sense here with " }{MPLTEXT 0 21 11 "scene=[x,y]" } {TEXT -1 34 ". Nevertheless, you must specify " }{MPLTEXT 0 21 11 "ar rows=none" }{TEXT -1 36 " to indicate that you don't want it." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "DEplot(\{diff(x(t),t)=1, di ff(y(t),t)=x(t)\}, [x(t),y(t)], t=0..1, [[x(0)=1,y(0)=1]], scene=[t,x] ,arrows=none);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "This bug has been corr ected in Release 5." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{MPLTEXT 0 21 6 "coords" }{TEXT -1 207 " option, which allows the use of non-Cart esian coordinate systems in many other plotting commands, does not wor k here. In two-dimensional plots it causes an error message. In thre e-dimensional plots (with " }{MPLTEXT 0 21 8 "DEplot3d" }{TEXT -1 75 " ) there is no error message, but the plot is done in Cartesian coordin ates." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "DEplot(\{diff(r(t) ,t)=-sin(theta(t)/2), diff(theta(t),t)=1/2\}, \n[r(t),theta(t)], t=0.. 3, \n[[r(0)=1,theta(0)=0]], arrows=none,coords=polar);" }}{PARA 8 "" 1 "" {TEXT -1 66 "Error, (in plot/options2d) unknown or bad argument, \+ coords = polar" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "DEplot3d (\{diff(r(t),t)=-sin(theta(t))/2, diff(theta(t),t)=1/2\}, \n[r(t),thet a(t)], t=-0..3, \n[[r(0)=1,theta(0)=0]], scene=[r,theta,t],\ncoords=cy lindrical);" }}}{PARA 0 "" 0 "" {TEXT -1 64 "A work-around is to produ ce a list of points, and plot it using " }{MPLTEXT 0 21 4 "plot" } {TEXT -1 24 " (in two dimensions) or " }{MPLTEXT 0 21 10 "spacecurve" }{TEXT -1 23 " (in three dimensions)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "L1:= op(indets(DEplot(\{diff(r(t),t)=-sin(theta(t)/2 ), diff(theta(t),t)=1/2\}, \n[r(t),theta(t)], t=0 .. 3, \n[[r(0)=1,the ta(0)=0]],arrows=none),list(list(numeric)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(L1,coords=polar,scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "L2:= op(indets(DEplot3d(\{d iff(r(t),t)=-sin(theta(t))/2, diff(theta(t),t)=1/2\}, \n[r(t),theta(t) ], t=-Pi..Pi, \n[[r(0)=1,theta(0)=0]], scene=[r,theta,t]),list(list(nu meric)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plots[spacecu rve](L2,coords=cylindrical);" }}}{PARA 0 "" 0 "" {TEXT -1 230 "This ca n be done for the direction field too (if you don't mind a bit of dist ortion in the arrows). In order to have the direction field and traje ctories in different colours, they should be plotted separately and co mbined using " }{MPLTEXT 0 21 7 "display" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "L3:= \{op(indets(DEplot(\{diff(r(t ),t)=-sin(theta(t)/2), diff(theta(t),t)=1/2\}, \n[r(t),theta(t)], t=0 \+ .. 3, r=0.1 .. 1.1,theta=0 .. 1.55, dirgrid=[10,8]),list(list(numeric) )))\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "p1:= plot(L1, co ords=polar, colour=blue):\np2:= plot(L3, coords=polar, colour=black): \nplots[display](\{p1,p2\},scaling=constrained,axes=box);" }}}{PARA 15 "" 0 "" {TEXT -1 259 "When the right side of one of the differentia l equations is identically 0, the direction field can not be plotted. \+ With no initial conditions, there is an error message. With initial \+ conditions, solution curves are plotted correctly but there are no arr ows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "DEplot(\{diff(x(t), t) = y(t)*x(t), diff(y(t),t) = 0\}, [x(t),y(t)], t=0..0.5, x=0..1, y=- 1..1, arrows=SMALL);\n" }}{PARA 8 "" 1 "" {TEXT -1 88 "Error, (in DEpl ot) Cannot produce plot, non-autonomous DE(s) require initial conditio ns." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "DEplot(\{diff(x(t), t) = y(t), diff(y(t),t) = 0\}, [x(t),y(t)], t=0 .. 0.5, \{[x(0)=.5,y( 0)=.5]\}, x=0..1, y=-1..1, arrows=SMALL);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "A work-around is to use a very small but nonzero value for the \+ derivative." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "DEplot(\{dif f(x(t),t) = y(t), diff(y(t),t) = 0.0001*y(t)\}, [x(t),y(t)], t=0 .. 0. 5, \{[x(0)=.5,y(0)=.5]\}, x=0..1, y=-1..1, arrows=SMALL);" }}}{PARA 15 "" 0 "" {TEXT -1 118 "The direction field plotter does not work for differential equations that are not of the form derivative = expressi on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "DEplot(diff(y(t),t) + y(t) = 1, y(t), t=0..1, [[y(0)=2]]);" }}}{PARA 0 "" 0 "" {TEXT -1 112 "The work-around is to rewrite the differential equations to put e verything except derivatives on the right side." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "DEplot(diff(y(t),t) = 1 - y(t), y(t), t=0..1, \+ [[y(0)=2]]);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "This bug has been correc ted in Release 5." }}{PARA 15 "" 0 "" {TEXT -1 66 "In some circumstanc es, the direction field plotter fails to do an " }{MPLTEXT 0 21 5 "eva lf" }{TEXT -1 86 ", with the result that some of the arrows to be plot ted contain symbolic expressions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "DEplot(diff(y(t),t) = exp(1-2*t), y(t), t = 0 .. 1,y= 0..1); " }}{PARA 8 "" 1 "" {TEXT -1 45 "Plotting error, non-numeric ve rtex definition" }}}{PARA 0 "" 0 "" {TEXT -1 87 "In this case, the pro blem doesn't occur if a floating-point value is placed inside the " } {MPLTEXT 0 21 3 "exp" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "DEplot(diff(y(t),t) = exp(1.0-2*t), y(t), t = 0 .. 1, y=0..1); " }}}{PARA 0 "" 0 "" {TEXT -1 30 "Another work-around is to m ap " }{MPLTEXT 0 21 5 "evalf" }{TEXT -1 50 " into every CURVES structu re in the DEplot result." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "pl:= DEplot(diff(y(t),t) = exp(1-2*t), y(t), t = 0 .. 1,y=0..1): \npt s:= indets(pl,specfunc(anything,CURVES)):\ns:= map(t -> (t = map(evalf ,t)), pts):\nsubs(s,pl);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "See \+ also:" }}{PARA 0 "" 0 "" {HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DEplot3d" 2 "DEplot3d" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " dfieldplot" 2 "dfieldplot" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "display " 2 "display" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "dsolve(numeric)" 2 "d solve,numeric" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fieldplot" 2 "fieldp lot" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "odeplot" 2 "odeplot" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "phaseportrait" 2 "phaseportrait" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "spacecurve" 2 "spacecurve" "" }{TEXT -1 1 " " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Maple Advisor Database " }{TEXT -1 14 " R. Israel 1998" }}}}{MARK "1 1 21" 1 }{VIEWOPTS 1 1 0 1 1 1803 }