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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 48 "Single First-order Differe
ntial Equation Example" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 18 "Initial
izing Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 71 "It is a good idea to re-initialize the Maple \"kernel\"
 at the beginning:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 47 "Now we load the Differential Equations package:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 21 "Drawing a Slope
 Field" }{TEXT 32 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "As an ex
ample, we consider the simple first-order differential equation  " }
{XPPEDIT 18 0 "diff(y(t),t) = t^2-y;" "6#/-%%diffG6$-%\"yG6#%\"tGF*,&*
$F*\"\"#\"\"\"F(!\"\"" }{TEXT -1 32 "  and assign it to the variable \+
" }{TEXT 0 2 "de" }{TEXT -1 137 " so that we don't have to write it ev
ery time.  Note that Maple \"keeps us honest\" about the unknown funct
ion -- we are required to enter " }{TEXT 0 4 "y(t)" }{TEXT -1 49 " ins
tead of the sloppy (but widespread) shortcut " }{TEXT 0 1 "y" }{TEXT 
-1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "de := diff(y(t),t) = t^2
 - y(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 148 "The following command displays (be patient!) the slope f
ield of the above differential equation in the appropriate ranges (the
 ranges are required):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dfieldplo
t( de, y(t), t=-2..2, y=-3..3 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "One can change the appearance of \+
the slope markers.  The following command displays the same slope fiel
d, but changes the slope markers from red arrows (default) to magenta \+
lines:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "dfieldplot( de, y
(t), t=-2..2, y=-3..3, arrows=LINE, color=magenta );" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 16 "Drawing Solutio
n" }{TEXT 32 1 "s" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 78 "The following command plots the solution that sati
sfies the initial condition " }{TEXT 263 1 "y" }{TEXT -1 36 "(-1) = 1,
 on top of the slope field:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "phas
eportrait( de, y(t), t=-2..2, [[y(-1)=1]] );" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Maple can plot more th
an one solution on the same plot.  The following command plots three s
olutions corresponding to the three initial conditions: " }{TEXT 264 
1 "y" }{TEXT -1 11 "(0) = 0.5, " }{TEXT 266 1 "y" }{TEXT -1 13 "(0) = \+
1, and " }{TEXT 267 1 "y" }{TEXT -1 147 "(0) = 1.5.  Note the brackets
 around each condition and around the entire list below (now you under
stand the double brackets in the example above)." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "phaseportrait( de, y(t), t=-2..2, [[y(0)=0.5], [y(0)=
1], [y(0)=1.5]] );" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 19 "Making It P
retty..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 254 "It is more convenient to save (in addition to the differ
ential equation itself) the t-range, the inititial conditions, and oth
er expressions for later use.  The following lines produce the same ou
tput as above, but are much easier to read and understand:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "trng := t=-2..2;\nics := [ [y(0)=0.
5], [y(0)=1], [y(0)=1.5] ];\nphaseportrait( de, y(t), trng, ics );" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "
One can change a lot of the features of the plot that Maple produces. \+
 The following command produces the same plot as above, but adds an ap
propriate title, plots the slopes as gray lines (instead of the defaul
t red arrows), and draws the solution curves in red, magenta, and maro
on (instead of the default gold color):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 124 "phaseportrait( de, y(t), trng, ics, title=\"Asymptotic soluti
ons\", arrows=LINE, color=gray, linecolor=[red,magenta,maroon] );" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 64 "To plot only the solution curves (without the slope field), use
 " }{TEXT 0 11 "arrows=NONE" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "phaseportrait( de, y(t), trng, ics, title=\"Asymptoti
c solutions\", arrows=NONE, linecolor=blue );" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Fo
r further information on DE plotting, execute the following help line:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?DEplot;" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "N
ow do it yourself by changing stuff around and enjoy the power of Mapl
e!" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 2 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }