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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 19 "Getting Started wit" }
{TEXT 256 0 "" }{TEXT -1 33 "h Differential Equations in Maple" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "In this Maple session, we see som
e of the basic tools for working with differential equations in Maple.
  We will only deal with first order equations here." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "First, we need to load \+
the \"DEtools\" library:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Let's define a
 differential equation.  As an example, we consider" }}{PARA 0 "" 0 "
" {TEXT -1 14 "   y' = 1-y^2." }}{PARA 0 "" 0 "" {TEXT -1 66 "This is \+
a first order, autonomous, ordinary differential equation." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following def
ines a variable called \"eq\" that holds the differential equation:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "eq := diff(y(t),t) = 1-y(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#eqG/-%%diffG6$-%\"yG6#%\"tGF,,&\"\"\"F.*$)F)\"\"#F.!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A few points:" }}{PARA 0 "" 0 "" 
{TEXT -1 63 "  1.  The derivative of y is specified with the \"diff\" \+
command." }}{PARA 0 "" 0 "" {TEXT -1 9 "  2.  We " }{TEXT 257 7 "can n
ot" }{TEXT -1 120 " drop the \"(t)\" from the dependent variable y.  M
aple treats \"y\" and \"y(t)\" differently, and our equation is for \"
y(t)\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
46 "Two useful commands are \"DEplot\" and \"dsolve\"." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 259 6 "DEplot" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 507 "Let's cr
eate a slope field (also known as a \"direction field\") for this equa
tion.  One Maple command that does this is called \"DEplot\".  In the \+
first example given here, the first argument is the differential equat
ion, the second is the function in the differential equation, the thir
d is the range the independent variable, and the fourth is the range o
f the dependent variable.  The option \"arrows=THIN\" tells the comman
d to show the slope field with thin arrows, and \"color=black\" makes \+
the arrows black." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 56 "DEplot(eq,y(t),t=-1..1,y=-3..3,arrows=THIN,c
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 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 320 "Let's add a solution curv
e to this plot.  To do this, we must specify an initial condition.  Le
t's try y(0)=0.  To specify initial conditions in DEplot, you put them
 in a list, contained in square brackets, just after the t range.  The
 added option \"linecolor=black\" tell the command to draw the solutio
n curve in black." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
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{PARA 0 "" 0 "" {TEXT -1 161 "Note that I used double square brackets \+
around the initial condition.  This is because you can enter several i
nitial conditions; each must be in square brackets:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 159 "DEplot(eq,y(t),t=-1..1,[[y(0)=-2],[y(0)=
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t$\"\"\"Fa]t7$F\\^tF]dt7$Fb^tF]dt7$Fg^tF]dt7$F\\_tF]dt7$Fa_tF]dt7$Ff_t
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F]dt777$%%FAILGFeetFdetFdetFdetFdetFdetFdet7$F[`t$\"3IH+jz[_#4%F*7$F``
t$\"3q#z$zDf1yHF*7$Fe`t$\"3w]RNU^RtBF*7$Fi`t$\"\"#Fa]t7$F[at$\"39MW90=
r]<F*7$F`at$\"31\">a_3&[v:F*7$Feat$\"3c=\"*z_b\"yW\"F*7$Fjat$\"3uQSDbI
M_8F*7$F_bt$\"3/YsTS!\\&z7F*7$Fdbt$\"3`O:rc6AB7F*7$Fibt$\"3q]_;w[8z6F*
7$F^ct$\"3+5SD13KW6F*7$Fcct$\"3f*z[BzMm6\"F*7$Fhct$\"3o^$=G*R\\%4\"F*7
77$Ff]t$!3o^$=G*R\\%4\"F*7$F\\^t$!3f*z[BzMm6\"F*7$Fb^t$!3+5SD13KW6F*7$
Fg^t$!3q]_;w[8z6F*7$F\\_t$!3`O:rc6AB7F*7$Fa_t$!3/YsTS!\\&z7F*7$Ff_t$!3
uQSDbIM_8F*7$F[`t$!3c=\"*z_b\"yW\"F*7$F``t$!31\">a_3&[v:F*7$Fe`t$!39MW
90=r]<F*7$Fi`t$!\"#Fa]t7$F[at$!3w]RNU^RtBF*7$F`at$!3q#z$zDf1yHF*7$Feat
$!3IH+jz[_#4%F*FdetFdetFdetFdetFdetFdetFdet777$Ff]t$!\"\"Fa]t7$F\\^tF]
[u7$Fb^tF][u7$Fg^tF][u7$F\\_tF][u7$Fa_tF][u7$Ff_tF][u7$F[`tF][u7$F``tF
][u7$Fe`tF][u7$Fi`tF][u7$F[atF][u7$F`atF][u7$FeatF][u7$FjatF][u7$F_btF
][u7$FdbtF][u7$FibtF][u7$F^ctF][u7$FcctF][u7$FhctF][u-%&STYLEG6#%%LINE
G-%*THICKNESSG6#\"\"$F]]t-%%VIEWG6$;$!++++D6Fh]t$\"++++D6Fh]t;$!++++vL
Fh]t$\"++++vLFh]t-%+AXESLABELSG6$Q\"t6\"Q%y(t)F\\^u-%&TITLEG6#%,Slope~
FieldG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Important Note! \+
 The solution curves plotted by the DEplot command have been computed \+
" }{TEXT 258 11 "numerically" }{TEXT -1 167 ",  using an algorithm sim
ilar to Euler's Method.  They are generally good approximations, but t
hey are not exact, and they may occasionally be very bad approximation
s." }}}{EXCHG {PARA 18 "" 0 "" {TEXT 260 18 "The Dsolve Command" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The command \"dso
lve\" can solve some differential equations " }{TEXT 261 12 "analytica
lly" }{TEXT -1 73 ".  Let's start out with a simple example that we ar
e quite familiar with:" }}{PARA 0 "" 0 "" {TEXT -1 11 "   y' = -y." }}
{PARA 0 "" 0 "" {TEXT -1 73 "First, we'll define the equation, and sav
e it in a variable called \"eq2\":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "eq2 := diff(y(t),t) = -y(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/-%%diffG6$-%\"yG6#%\"tGF,,$F)!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 147 "Now we'll use \"dsolve\" to find the gen
eral solution.  The first argument of \"dsolve\" is the equation, and \+
the second is the function  to solve for:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "dsolve(eq2,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%\"yG6#%\"tG*&%$_C1G\"\"\"-%$expG6#,$F'!\"\"F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 211 "This should look familiar!  Notice that Maple ha
s called the arbitrary constant \"_C1\".  This is a common Maple conve
ntion; when it creates a constant, it typically put an underscore in t
he beginning of the name." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 81 "We can also use \"dsolve\" to solve an initial val
ue problem.  For example, suppose" }}{PARA 0 "" 0 "" {TEXT -1 9 "  y(0
)=2." }}{PARA 0 "" 0 "" {TEXT -1 158 "We specify this in the command b
y combining the differential equation with the initial condition in \"
curly brackets\", thus creating the initial value problem:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(\{eq2,y(0)=2\},y(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$-%$expG6#,$F'!\"\"\"\"
#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "Look at that example carefu
lly.  The first argument is \{eq2,y(0)=2\}.  In Maple, curly brackets \+
are commonly used to create lists of things.  In this case, we make a \+
 list of the equation and the initial condition, and pass this to \"ds
olve\" as its first argument.  The second argument, y(t), stays the sa
me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 480 "H
ow do we plot this solution?  Unfortunately, the solution given above \+
is not in a format that is easily plotted by the \"plot\" command.  Th
e plot command plots expressions, but the above solution is actually a
n equation (it has an equals sign sitting the middle).  The expression
 that we want, namely 2*exp(-t), is the right-hand-side of the equatio
n.  Fortunately, there is a command in Maple that lets us take the rig
ht-hand-side from an equation.  The function is called \"rhs\"." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "So here \+
is one way to plot the solution above.  First, save the solution in a \+
variable, say \"sol2\" (because it solves eq2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "sol2 := dsolve(\{eq2,y(0)=2\},y(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/-%\"yG6#%\"tG,$-%$expG6#,$F)!\"\"\"
\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Now take out the right-han
d-side, and save it in a new variable called rhsSol2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rhsSol2 := rhs(sol2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(rhsSol2G,$-%$expG6#,$%\"tG!\"\"\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Now plot it:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "plot(rhsSol2,t=-1..2);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$\"3=!
4=pljlV&!#<7$$!3[*****\\P&3Y$*!#=$\"3QMInGIV#4&F-7$$!3C++Dcx6x()F1$\"3
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3+VL<([)RRPF-7$$!3j***\\i+#QUcF1$\"31jo\"Q*e@;NF-7$$!3$****\\i!3%f+&F1
$\"3gYv7q?S*H$F-7$$!3;++D\"oS:P%F1$\"33G^0.#*e'4$F-7$$!3h*****\\<#)*=P
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\\-\\r\\#F1$\"3UL%fGz=tc#F-7$$!3?+++vGVZ=F1$\"3'=l9r>>eS#F-7$$!3_*****
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)***\\P/QBE\"F1$\"3/8'R4\\<Gw\"F-7$$\"39******\\\"o?&=F1$\"3%GB&eL['=m
\"F-7$$\"3k++vVb4*\\#F1$\"3)G1h2XUxb\"F-7$$\"3w++DJ'=_6$F1$\"3i\"4Y&)>
jYY\"F-7$$\"3#4++vVy!ePF1$\"3kkuHh&oMP\"F-7$$\"3'4+](=WU[VF1$\"3kA'RQJ
LZH\"F-7$$\"3s****\\7B>&)\\F1$\"3/i3I8*e[@\"F-7$$\"3w***\\P>:mk&F1$\"3
3dVV;^5P6F-7$$\"3d***\\iv&QAiF1$\"3C%3E.6$[t5F-7$$\"3j++]PPBWoF1$\"32j
#3,(>w35F-7$$\"3%*)*****\\Nm'[(F1$\"3o)*\\$f%)Q*f%*F17$$\"36****\\(yb^
6)F1$\"3GCg&*)=(o$)))F17$$\"3')***\\PMaKs)F1$\"3s*zKhto&f$)F17$$\"3a**
**\\7TW)R*F1$\"3zN.nZBx8yF17$$\"3z*****\\@80+\"F-$\"3C89FnP\"QN(F17$$
\"31++]7,Hl5F-$\"3OG@E)=dD*oF17$$\"3()**\\P4w)R7\"F-$\"3sgH0yhi*\\'F17
$$\"3;++]x%f\")=\"F-$\"3o3\\&[6Mc4'F17$$\"3!)**\\P/-a[7F-$\"3W@Be9oYQd
F17$$\"3/+](=Yb;J\"F-$\"3')\\-&=)\\Z(Q&F17$$\"3')****\\i@Ot8F-$\"3Yb3o
043l]F17$$\"3')**\\PfL'zV\"F-$\"3eKk#=u:#[ZF17$$\"3>+++!*>=+:F-$\"3W<(
3x7\"zhWF17$$\"3-++DE&4Qc\"F-$\"3/$p<P]Tn=%F17$$\"3=+]P%>5pi\"F-$\"3X/
7-$*yrIRF17$$\"39+++bJ*[o\"F-$\"3%Q)o\\O\\G4PF17$$\"33++Dr\"[8v\"F-$\"
3SXWN&['zqMF17$$\"3++++Ijy5=F-$\"3eC*QX-50F$F17$$\"31+]P/)fT(=F-$\"3+L
GA$Qx'pIF17$$\"31+]i0j\"[$>F-$\"3AKPsBg,*)GF17$$\"\"#F*$\"30aAtkcq1FF1
-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%+AXESLABELSG6$Q\"t6\"Q!Ff[l-%%VIEW
G6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "You ca
n combine several of those steps:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "sol2 := rhs(dsolve(\{eq2,y(0)=2\},y(t)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%sol2G,$-%$expG6#,$%\"tG!\"\"\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(sol2,t=-1..2);" }}
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%>5pi\"F-$\"3X/7-$*yrIRF17$$\"39+++bJ*[o\"F-$\"3%Q)o\\O\\G4PF17$$\"33+
+Dr\"[8v\"F-$\"3SXWN&['zqMF17$$\"3++++Ijy5=F-$\"3eC*QX-50F$F17$$\"31+]
P/)fT(=F-$\"3+LGA$Qx'pIF17$$\"31+]i0j\"[$>F-$\"3AKPsBg,*)GF17$$\"\"#F*
$\"30aAtkcq1FF1-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%+AXESLABELSG6$Q\"t6
\"Q!Ff[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
62 "Let's try another example.  Recall that \"eq\" is still defined:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "eq;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"tGF*,&\"\"\"F,*$)F'\"\"#F,!\"\"" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Let's give this to \"dsolve\":
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol1 := rhs(dsolve(eq,y
(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G-%%tanhG6#,&%\"tG\"
\"\"%$_C1GF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "So the solution \+
is the hyperbolic tangent function.  Does this really give us all the \+
solutions that we expect?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 67 "Let's try some of the same initial conditions that
 we used earlier:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol1a \+
:= rhs(dsolve(\{eq,y(0)=-2\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&sol1aG-%%cothG6#,&%\"tG\"\"\"*&#F*\"\"#F*-%#lnG6#\"\"$F*!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sol1a,t=-1..1);" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7`p7
$$!\"\"\"\"!$!3m(3[(\\f[%4\"!#<7$$!3ommm;p0k&*!#=$!3)GJC6[RN5\"F-7$$!3
wKL$3<XZ=*F1$!3-G!G:Zd@6\"F-7$$!3mmmmT%p\"e()F1$!3()>_l[$fF7\"F-7$$!3:
mmm\"4m(G$)F1$!3eyL\"fK3X8\"F-7$$!3\"QLL3i.9!zF1$!3!4X@rE&RZ6F-7$$!3\"
ommT!R=0vF1$!35+7FYj_g6F-7$$!3u****\\P8#\\4(F1$!3%)41YkwXv6F-7$$!3+nm;
/siqmF1$!35Q@gpB\\#>\"F-7$$!3[++](y$pZiF1$!31\"y&*R,y7@\"F-7$$!33LLL$y
aE\"eF1$!3IlMw9\">FB\"F-7$$!3hmmm\">s%HaF1$!3c=ft0bg`7F-7$$!3Q+++]$*4)
*\\F1$!3!=>5J)>lz7F-7$$!39+++]_&\\c%F1$!3K6%QUKm)38F-7$$!31+++]1aZTF1$
!3<&*[n3WLS8F-7$$!3umm;/#)[oPF1$!3-zf'QvD@P\"F-7$$!3hLLL$=exJ$F1$!3bb!
QqI0XT\"F-7$$!3*RLLLtIf$HF1$!3=1%\\2g!)[X\"F-7$$!3]++]PYx\"\\#F1$!3`L>
ppw'y]\"F-7$$!3EMLLL7i)4#F1$!3o\">lHro5c\"F-7$$!3c****\\P'psm\"F1$!3$*
3mrT?[F;F-7$$!3')****\\74_c7F1$!3yysM#o4+q\"F-7$$!3)3LLL3x%z#)!#>$!3[u
,$>;>ty\"F-7$$!3KMLL3s$QM%Fer$!3cToIG1<!)=F-7$$!3]^omm;zr)*!#@$!3s))G2
*HWq*>F-7$$\"3%pJL$ezw5VFer$!3Y8>7y(G99#F-7$$\"3s*)***\\PQ#\\\")Fer$!3
g\"**4T)>J\"H#F-7$$\"3GKLLe\"*[H7F1$!3o&fV54xe[#F-7$$\"3I*******pvxl\"
F1$!3he?$)fT(Rt#F-7$$\"3#z****\\_qn2#F1$!3TtlkV7:SIF-7$$\"3U)***\\i&p@
[#F1$!3@s5S;n.@MF-7$$\"3B)****\\2'HKHF1$!3=p)[!*[q+*RF-7$$\"3ElmmmZvOL
F1$!3Or.BJV@4ZF-7$$\"3i******\\2goPF1$!3)41`q\"GGceF-7$$\"3UKL$eR<*fTF
1$!3ax:Ob/XXvF-7$$\"3/mm\"HiBQP%F1$!3ALoqA%G>(*)F-7$$\"3m******\\)Hxe%
F1$!3^ugw%y$e26!#;7$$\"37KLeR*)**)y%F1$!3hzH=VXnA9F_w7$$\"3ckm;H!o-*\\
F1$!3[K.W<Tc!*>F_w7$$\"3c(****\\7ga4&F1$!3Ccg&>S2k^#F_w7$$\"3nJL$3A_1?
&F1$!3eCm1<%R3U$F_w7$$\"3G***\\(o#[KD&F1$!3?(\\#zO?rqTF_w7$$\"3ylmm;V%
eI&F1$!3b!z9xq:?M&F_w7$$\"3.**\\iSB9K`F1$!3==Qs(**Q[@'F_w7$$\"3GKLek.W
e`F1$!3#*okq\\fqGuF_w7$$\"3!*)\\ilP*er`F1$!3t3M>>-vK#)F_w7$$\"3_l;a)QQ
ZQ&F1$!3&)ef,#>1?B*F_w7$$\"39K3_+u)yR&F1$!3Mc=mx$Q20\"!#:7$$\"3y)***\\
7k.6aF1$!3.$f8\"*Qn\">7Fgz7$$\"3F2-8v>*QU&F1$!3E2B'4jZdW\"Fgz7$$\"3v:/
wPvuOaF1$!3g(\\O\"4\"zdx\"Fgz7$$\"3CC1R+Jg\\aF1$!3T,)fpio5I#Fgz7$$\"3s
K3-j'eCY&F1$!3Y6YgbronKFgz7$$\"3?T5lDUJvaF1$!3]44/p>pMcFgz7$$\"3o\\7G)
yp\")[&F1$!3#z=t#H!)HW?!#97$$\"31d9\"4ND5]&F1$\"3U80Yl_mb7F[]l7$$\"3al
;a84)Q^&F1$\"345wKKJ?.[Fgz7$$\"3-u=<wktEbF1$\"3]#fZ%>.dpHFgz7$$\"3S\"3
-)Q?fRbF1$\"3ok?P#4R\"\\@Fgz7$$\"3))*GK9gZCb&F1$\"3GA!)euQ\"Ro\"Fgz7$$
\"3O)\\iS;.`c&F1$\"31Z'4'[dE%Q\"Fgz7$$\"3'oq#pE(e\"ybF1$\"37.vx5S:v6Fg
z7$$\"3M:HK*G95f&F1$\"3#[NpM.K4-\"Fgz7$$\"3#Q7`>&)pQg&F1$\"3y@>fS_&\\-
*F_w7$$\"3IKLe9as;cF1$\"3%Q^g\"H=%o3)F_w7$$\"3E\\P%)RlVUcF1$\"3g?I/aU0
&p'F_w7$$\"3AmT5lw9ocF1$\"3mBl-0`07dF_w7$$\"3?$ek.zeQp&F1$\"3CrVh&y93)
\\F_w7$$\"31**\\i:*p&>dF1$\"3&*G8i,4g:WF_w7$$\"3)=$e9m@*4x&F1$\"3=IhE6
`%*)f$F_w7$$\"3#emmmT9C#eF1$\"3$*=!\\=%RNPIF_w7$$\"3WKLeR<vPgF1$\"3_F'
3$R4sP=F_w7$$\"33****\\i!*3`iF1$\"3D%eU'3TF=8F_w7$$\"3Ymm\"z\\%[gkF1$
\"3Y2)zFn'*o.\"F_w7$$\"3%QLLL$*zym'F1$\"3W@2;Ey2^&)F-7$$\"3ILL$3sr*zoF
1$\"3]`*oA#HWcsF-7$$\"3wKLL3N1#4(F1$\"3yyt&R05rI'F-7$$\"3Nmm;HYt7vF1$
\"3,66%y=O%=]F-7$$\"3Y*******p(G**yF1$\"3>KXi0[yNUF-7$$\"3]mmmT6KU$)F1
$\"3C-^\"*G)[Tg$F-7$$\"3fKLLLbdQ()F1$\"31M,P9wg)=$F-7$$\"3[++]i`1h\"*F
1$\"3mpf3ICYZGF-7$$\"3W++]P?Wl&*F1$\"3c5(p*pY$)*e#F-7$$\"\"\"F*$\"3U;o
cDl.nBF--%'COLOURG6&%$RGBG$\"#5F)$F*F*Fcfl-%+AXESLABELSG6$Q\"t6\"Q!Fhf
l-%%VIEWG6$;F(Fiel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
172 "What is going on here?  This doesn't look like anything we saw wi
th the \"DEplot\" command.  Let's restrict the y range to be the same \+
as the slope field plots shown earlier:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "plot(sol1a,t=-1..1,y=-3..3);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7`p7$$!\"\"\"\"!$!3m(
3[(\\f[%4\"!#<7$$!3ommm;p0k&*!#=$!3)GJC6[RN5\"F-7$$!3wKL$3<XZ=*F1$!3-G
!G:Zd@6\"F-7$$!3mmmmT%p\"e()F1$!3()>_l[$fF7\"F-7$$!3:mmm\"4m(G$)F1$!3e
yL\"fK3X8\"F-7$$!3\"QLL3i.9!zF1$!3!4X@rE&RZ6F-7$$!3\"ommT!R=0vF1$!35+7
FYj_g6F-7$$!3u****\\P8#\\4(F1$!3%)41YkwXv6F-7$$!3+nm;/siqmF1$!35Q@gpB
\\#>\"F-7$$!3[++](y$pZiF1$!31\"y&*R,y7@\"F-7$$!33LLL$yaE\"eF1$!3IlMw9
\">FB\"F-7$$!3hmmm\">s%HaF1$!3c=ft0bg`7F-7$$!3Q+++]$*4)*\\F1$!3!=>5J)>
lz7F-7$$!39+++]_&\\c%F1$!3K6%QUKm)38F-7$$!31+++]1aZTF1$!3<&*[n3WLS8F-7
$$!3umm;/#)[oPF1$!3-zf'QvD@P\"F-7$$!3hLLL$=exJ$F1$!3bb!QqI0XT\"F-7$$!3
*RLLLtIf$HF1$!3=1%\\2g!)[X\"F-7$$!3]++]PYx\"\\#F1$!3`L>ppw'y]\"F-7$$!3
EMLLL7i)4#F1$!3o\">lHro5c\"F-7$$!3c****\\P'psm\"F1$!3$*3mrT?[F;F-7$$!3
')****\\74_c7F1$!3yysM#o4+q\"F-7$$!3)3LLL3x%z#)!#>$!3[u,$>;>ty\"F-7$$!
3KMLL3s$QM%Fer$!3cToIG1<!)=F-7$$!3]^omm;zr)*!#@$!3s))G2*HWq*>F-7$$\"3%
pJL$ezw5VFer$!3Y8>7y(G99#F-7$$\"3s*)***\\PQ#\\\")Fer$!3g\"**4T)>J\"H#F
-7$$\"3GKLLe\"*[H7F1$!3o&fV54xe[#F-7$$\"3I*******pvxl\"F1$!3he?$)fT(Rt
#F-7$$\"3#z****\\_qn2#F1$!3TtlkV7:SIF-7$$\"3U)***\\i&p@[#F1$!3@s5S;n.@
MF-7$$\"3B)****\\2'HKHF1$!3=p)[!*[q+*RF-7$$\"3ElmmmZvOLF1$!3Or.BJV@4ZF
-7$$\"3i******\\2goPF1$!3)41`q\"GGceF-7$$\"3UKL$eR<*fTF1$!3ax:Ob/XXvF-
7$$\"3/mm\"HiBQP%F1$!3ALoqA%G>(*)F-7$$\"3m******\\)Hxe%F1$!3^ugw%y$e26
!#;7$$\"37KLeR*)**)y%F1$!3hzH=VXnA9F_w7$$\"3ckm;H!o-*\\F1$!3[K.W<Tc!*>
F_w7$$\"3c(****\\7ga4&F1$!3Ccg&>S2k^#F_w7$$\"3nJL$3A_1?&F1$!3eCm1<%R3U
$F_w7$$\"3G***\\(o#[KD&F1$!3?(\\#zO?rqTF_w7$$\"3ylmm;V%eI&F1$!3b!z9xq:
?M&F_w7$$\"3.**\\iSB9K`F1$!3==Qs(**Q[@'F_w7$$\"3GKLek.We`F1$!3#*okq\\f
qGuF_w7$$\"3!*)\\ilP*er`F1$!3t3M>>-vK#)F_w7$$\"3_l;a)QQZQ&F1$!3&)ef,#>
1?B*F_w7$$\"39K3_+u)yR&F1$!3Mc=mx$Q20\"!#:7$$\"3y)***\\7k.6aF1$!3.$f8
\"*Qn\">7Fgz7$$\"3F2-8v>*QU&F1$!3E2B'4jZdW\"Fgz7$$\"3v:/wPvuOaF1$!3g(
\\O\"4\"zdx\"Fgz7$$\"3CC1R+Jg\\aF1$!3T,)fpio5I#Fgz7$$\"3sK3-j'eCY&F1$!
3Y6YgbronKFgz7$$\"3?T5lDUJvaF1$!3]44/p>pMcFgz7$$\"3o\\7G)yp\")[&F1$!3#
z=t#H!)HW?!#97$$\"31d9\"4ND5]&F1$\"3U80Yl_mb7F[]l7$$\"3al;a84)Q^&F1$\"
345wKKJ?.[Fgz7$$\"3-u=<wktEbF1$\"3]#fZ%>.dpHFgz7$$\"3S\"3-)Q?fRbF1$\"3
ok?P#4R\"\\@Fgz7$$\"3))*GK9gZCb&F1$\"3GA!)euQ\"Ro\"Fgz7$$\"3O)\\iS;.`c
&F1$\"31Z'4'[dE%Q\"Fgz7$$\"3'oq#pE(e\"ybF1$\"37.vx5S:v6Fgz7$$\"3M:HK*G
95f&F1$\"3#[NpM.K4-\"Fgz7$$\"3#Q7`>&)pQg&F1$\"3y@>fS_&\\-*F_w7$$\"3IKL
e9as;cF1$\"3%Q^g\"H=%o3)F_w7$$\"3E\\P%)RlVUcF1$\"3g?I/aU0&p'F_w7$$\"3A
mT5lw9ocF1$\"3mBl-0`07dF_w7$$\"3?$ek.zeQp&F1$\"3CrVh&y93)\\F_w7$$\"31*
*\\i:*p&>dF1$\"3&*G8i,4g:WF_w7$$\"3)=$e9m@*4x&F1$\"3=IhE6`%*)f$F_w7$$
\"3#emmmT9C#eF1$\"3$*=!\\=%RNPIF_w7$$\"3WKLeR<vPgF1$\"3_F'3$R4sP=F_w7$
$\"33****\\i!*3`iF1$\"3D%eU'3TF=8F_w7$$\"3Ymm\"z\\%[gkF1$\"3Y2)zFn'*o.
\"F_w7$$\"3%QLLL$*zym'F1$\"3W@2;Ey2^&)F-7$$\"3ILL$3sr*zoF1$\"3]`*oA#HW
csF-7$$\"3wKLL3N1#4(F1$\"3yyt&R05rI'F-7$$\"3Nmm;HYt7vF1$\"3,66%y=O%=]F
-7$$\"3Y*******p(G**yF1$\"3>KXi0[yNUF-7$$\"3]mmmT6KU$)F1$\"3C-^\"*G)[T
g$F-7$$\"3fKLLLbdQ()F1$\"31M,P9wg)=$F-7$$\"3[++]i`1h\"*F1$\"3mpf3ICYZG
F-7$$\"3W++]P?Wl&*F1$\"3c5(p*pY$)*e#F-7$$\"\"\"F*$\"3U;ocDl.nBF--%'COL
OURG6&%$RGBG$\"#5F)$F*F*Fcfl-%+AXESLABELSG6$Q\"t6\"Q\"yFhfl-%%VIEWG6$;
F(Fiel;$!\"$F*$\"\"$F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 260 "Aha! \+
 We see the solution where y(0)=-2.  What is happening is that this so
lution \"blows up\" in finite time.  Maple doesn't know that we don't \+
care about what happens after this point, so it also plots the functio
n for values of t after the vertical asymptote." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Let's try the initial con
dition y(0)=0.  This one shouldn't have any vertical asymptotes." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "sol1b := rhs(dsolve(\{eq,y(0)=0\},y(t)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sol1bG-%%tanhG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "plot(sol1b,t=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 
400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$!3^[wbf:%fh(!#=
7$$!3ommm;p0k&*F-$!3j<GIcKpEuF-7$$!3wKL$3<XZ=*F-$!31)=+q]\\<D(F-7$$!3m
mmmT%p\"e()F-$!3XK\"e5HvJ/(F-7$$!3:mmm\"4m(G$)F-$!3vqy$['e<?oF-7$$!3\"
QLL3i.9!zF-$!3%\\2@QV&)[e'F-7$$!3\"ommT!R=0vF-$!3SA(>$\\6eajF-7$$!3u**
**\\P8#\\4(F-$!3\\$*[?WEe.hF-7$$!3+nm;/siqmF-$!39*z^jCW/$eF-7$$!3[++](
y$pZiF-$!3zT'eN:+Wa&F-7$$!33LLL$yaE\"eF-$!3k;$QF(f%eB&F-7$$!3hmmm\">s%
HaF-$!3U7Wx*ec@&\\F-7$$!3Q+++]$*4)*\\F-$!3kNXVzmn>YF-7$$!39+++]_&\\c%F
-$!3+Dg)*ymBsUF-7$$!31+++]1aZTF-$!3E[P5H$=]#RF-7$$!3umm;/#)[oPF-$!3)=K
cYCx'*f$F-7$$!3hLLL$=exJ$F-$!3\")y\\.]_:,KF-7$$!3*RLLLtIf$HF-$!3/?B&*Q
^QaGF-7$$!3]++]PYx\"\\#F-$!3h,B%\\5`9W#F-7$$!3EMLLL7i)4#F-$!3cj$GEDX$o
?F-7$$!3c****\\P'psm\"F-$!3/)R1/l!*>l\"F-7$$!3')****\\74_c7F-$!3#3\"[^
r&\\*\\7F-7$$!3)3LLL3x%z#)!#>$!3AJ\"H1G51E)Fdr7$$!3KMLL3s$QM%Fdr$!3B%4
t8:26M%Fdr7$$!3]^omm;zr)*!#@$!3'))Q1*f%)yr)*F_s7$$\"3%pJL$ezw5VFdr$\"3
Oi<V[(*43VFdr7$$\"3s*)***\\PQ#\\\")Fdr$\"3^')QdYkCJ\")Fdr7$$\"3GKLLe\"
*[H7F-$\"3&Qpo`AJLA\"F-7$$\"3I*******pvxl\"F-$\"3\"\\6nNOaFk\"F-7$$\"3
#z****\\_qn2#F-$\"3/v.60*>u/#F-7$$\"3U)***\\i&p@[#F-$\"33V-7V%=CV#F-7$
$\"3B)****\\2'HKHF-$\"3siKooi/^GF-7$$\"3ElmmmZvOLF-$\"3U.F.'z%>=KF-7$$
\"3i******\\2goPF-$\"3SW!on?v(*f$F-7$$\"3UKL$eR<*fTF-$\"3faX)HD$[NRF-7
$$\"3m******\\)Hxe%F-$\"3OE'Q#Gj$3H%F-7$$\"3ckm;H!o-*\\F-$\"3?=,(\\W9N
h%F-7$$\"3y)***\\7k.6aF-$\"37*3:\\FH#Q\\F-7$$\"3#emmmT9C#eF-$\"3$*ycqr
k#HC&F-7$$\"33****\\i!*3`iF-$\"3?,N2$yN\"[bF-7$$\"3%QLLL$*zym'F-$\"3b9
Z]%fI'GeF-7$$\"3wKLL3N1#4(F-$\"3vC]&*\\\"*y,hF-7$$\"3Nmm;HYt7vF-$\"3C#
\\%yy13fjF-7$$\"3Y*******p(G**yF-$\"3?\"HHW#oo$e'F-7$$\"3]mmmT6KU$)F-$
\"3;^1%*4\">u#oF-7$$\"3fKLLLbdQ()F-$\"3cy5]mvGLqF-7$$\"3[++]i`1h\"*F-$
\"3qZlT9J]SsF-7$$\"3W++]P?Wl&*F-$\"3([VV\"RVJFuF-7$$\"\"\"F*$\"3^[wbf:
%fh(F--%'COLOURG6&%$RGBG$\"#5F)$F*F*F`[l-%+AXESLABELSG6$Q\"t6\"Q!Fe[l-
%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "This \+
one is well-behaved.  If we use a longer range of time, we can clearly
 see what happens as t becomes very large or as -t becomes very large:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sol1b,t=-4..4);" }
}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7
$$!\"%\"\"!$!3Mq1R(*HH$***!#=7$$!3ommmmFiDQ!#<$!3-q%\\v[&\\!***F-7$$!3
5LLLo!)*Qn$F1$!3;!4'[(3Gr)**F-7$$!3nmmmwxE.NF1$!391e&[c(*=)**F-7$$!3Ym
mmOk]JLF1$!3YOV$Hu'[u**F-7$$!3_LLL[9cgJF1$!3-_!\\Z11T'**F-7$$!3smmmhN2
-IF1$!3#)z5q$p^2&**F-7$$!3!******\\`oz$GF1$!3>dOUK]oJ**F-7$$!3!omm;)3D
oEF1$!3-e5*[Z2U!**F-7$$!3?+++:v2*\\#F1$!3OjK\")[u*e')*F-7$$!3BLLL8>1DB
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[l-%+AXESLABELSG6$Q\"t6\"Q!Ff[l-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 98 "Finally, let's put an initial condition i
nto \"dsolve\" that should give us an equilibrium solution:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol1c := rhs(dsolve(\{eq,y(0
)=1\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol1cG\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "That worked as expected." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 232 "Now, recall th
at the general solution given by dsolve was tanh(t+_C1).  Unfortunatel
y, this is only the correct solution if -1 < y(0) < 1.  This is a exam
ple where we must be aware of Maple's quirks if we are to use it succe
ssfully." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 101 "Can Maple solve any first order differential equation?  \+
Unfortunately, no.  Here is a simple example:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "eq3 := diff(y(t),t) = sin(y(t))+cos(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/-%%diffG6$-%\"yG6#%\"tGF,,&-%$
sinG6#F)\"\"\"-%$cosGF+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "dsolve(eq3,y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The func
tion returns nothing; Maple can not solve this equation." }}}}{MARK "0
 0 2" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 
}