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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 40 "                        \+
                " }{TEXT 256 29 "HOW TO DRAW A DIRECTION FIELD" }
{TEXT -1 29 "                             " }{TEXT 276 4 "FOM1" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }
{TEXT 261 15 "direction field" }{TEXT -1 145 " is a collection of line
 segments (or vectors) that are tangent to solutions of a differential
 equation.  For a first order differential equation" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "                     " 
}{TEXT 257 14 "dy/dt = f(t,y)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 13 "the function " }{TEXT 258 6 "f(t,y)" }
{TEXT 260 63 " is the slope of the tangent to the solution at the poin
t (t,y)" }{TEXT -1 46 ".  To draw a direction field we must evaluate \+
" }{TEXT 259 6 "f(t,y)" }{TEXT -1 164 " at a large number of points an
d then draw a line segment at each point having the corresponding slop
e.  This is very tedious to do by hand, but is easy with Maple." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 4 "NOTE" }
{TEXT -1 13 " that you do " }{TEXT 262 3 "not" }{TEXT -1 9 " have to \+
" }{TEXT 264 5 "solve" }{TEXT -1 170 " the differential equation in or
der to draw a direction field.  Consequently, it is possible to draw d
irection fields fairly easily even for quite complicted equations.  " 
}{TEXT 263 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 121 "Once a direction field has been drawn, i
t is usually possible to visualize the general qualitative behavior of
 solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
265 8 "EXAMPLE " }{TEXT -1 41 "  Draw a direction field for the equati
on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "   \+
                 " }{TEXT 266 6 "dy/dt " }{TEXT -1 1 "=" }{TEXT 267 
12 " 2 - y - t/2" }{TEXT -1 6 " .    " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 268 10 "SOLUTION  " }{TEXT -1 20 "We use the
 command  " }{TEXT 269 6 "DEplot" }{TEXT -1 77 "  to draw the directio
n field.  This command is located in a package called  " }{TEXT 270 7 
"DEtools" }{TEXT -1 36 " , so we first call up this package." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "w
ith(DEtools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7aq%)DEnormalG%'DEplo
tG%)DEplot3dG%/DEplot_polygonG%(DFactorG%,DFactorLCLMG%,DFactorsolsG%+
DchangevarG%%GCRDG%%LCLMG%0PDEchangecoordsG%-RiemannPsolsG%(XchangeG%,
XcommutatorG%'XgaugeG%(abelsolG%(adjointG%+autonomousG%-bernoullisolG%
)buildsolG%)buildsymG%'canoniG%)caseplotG%*casesplitG%*checkrankG%)chi
nisolG%,clairautsolG%/constcoeffsolsG%+convertAlgG%+convertsysG%-dalem
bertsolG%(dcoeffsG%*de2diffopG%+dfieldplotG%*diffop2deG%&dsubsG%*eigen
ringG%6endomorphism_charpolyG%'equinvG%&eta_kG%*eulersolsG%)exactsolG%
(expsolsG%/exterior_powerG%'firintG%(firtestG%+formal_solG%(gen_expG%,
generate_icG%+genhomosolG%'gensysG%-hamilton_eqsG%+indicialeqG%'infgen
G%,initialdataG%/integrate_solsG%*intfactorG%+invariantsG%,kovacicsols
G%-leftdivisionG%'liesolG%)line_intG%*linearsolG%)matrixDEG%/matrix_ri
ccatiG%-moser_reduceG%)muchangeG%%multG%'mutestG%/newton_polygonG%)nor
malG2G%+odeadvisorG%'odepdeG%.parametricsolG%.phaseportraitG%)poincare
G%)polysolsG%(ratsolsG%'redodeG%,reduceOrderG%-reduce_orderG%.regular_
partsG%*regularspG%.remove_RootOfG%/riccati_systemG%+riccatisolG%(rifs
impG%.rightdivisionG%(rtaylorG%-separablesolG%,solve_groupG%-super_red
uceG%'symgenG%0symmetric_powerG%2symmetric_productG%(symtestG%)transin
vG%*translateG%,untranslateG%)varparamG%%zoomG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "Maple's response lists the (many!) commands in the \+
" }{TEXT 271 8 "DEtools " }{TEXT -1 79 "package.  [You can suppress th
is list by using a colon instead of a semicolon.]" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Next we enter the given d
ifferential equation.  " }{TEXT 274 4 "NOTE" }{TEXT -1 141 " that the \+
dependent variable y is written as a function of the independent varia
ble t, which means y(t), wherever it appears in the equation." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "eq:=diff(y(t),t) = 2 - y(t) -t/2;" }}{PARA 11 "" 1 "" {XPPMATH 
275 "6#>%#eqG/-%%diffG6$-%\"yG6#%\"tGF,,(\"\"#\"\"\"F)!\"\"*&#F/F.F/F,
F/F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 361 "The next command draws the direction field.  The first argumen
t specifies the equation, the second argument gives the dependent vari
able as a function of the independent variable, the third and fourth a
rguments define the rectangle on which the plot is to be drawn, and th
e fifth argument specifies the type of arrows to be used (small, mediu
m, large, or line)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "DEplot(eq, y(t), t=0..10, y =-5..5, arrow
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t solutions appear to approach a line (or perhaps a curve) that has ne
gative slope and crosses the t-axis near t = 5." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "More detailed information
 requires further investigation." }}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 
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