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{SECT 0 {PARA 259 "" 0 "" {TEXT 324 38 "Maple Tools for Differential E
quations" }}{PARA 258 "" 0 "" {TEXT 325 0 "" }{TEXT -1 10 "A. J. Meir
" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 46 "C
opyright (C) A. J. Meir. All rights reserved." }}{PARA 257 "" 0 "" 
{TEXT -1 492 "This worksheet is for educational use only. No part of t
his publication may be reproduced or transmitted for profit in any for
m or by any means, electronic or mechanical, including photocopy, reco
rding, or any information storage and retrieval system without prior w
ritten permission from the author. Not for profit distribution of the \+
software is allowed without prior written permission, providing that t
he worksheet is not modified in any way and full credit to the author \+
is acknowledged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 326 91 "(This document is largely based on a similar document or
iginally written by P. G. Schmidt.)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Maple will find " }{TEXT 
256 18 "symbolic solutions" }{TEXT -1 11 " (that is, " }{TEXT 257 21 "
closed-form solutions" }{TEXT -1 4 " or " }{TEXT 258 17 "solution form
ulas" }{TEXT -1 131 ") for many types of d.e.s and i.v.p.s, including \+
separable and linear first-order equations. Moreover, Maple can comput
e and graph " }{TEXT 259 19 "numerical solutions" }{TEXT -1 11 " (that
 is, " }{TEXT 260 21 "approximate solutions" }{TEXT -1 50 ") for virtu
ally all i.v.p.s of practical interest." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 261 43 "1. Finding and Plotting Symbolic So
lutions." }{TEXT -1 60 "  To find a general solution for a given d.e.,
 use Maple's  " }{TEXT 262 6 "dsolve" }{TEXT -1 34 "  command, whose g
eneral format is" }}{PARA 256 "" 0 "" {TEXT 263 27 "dsolve(DE,unknown_
function)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "The followin
g command, for example, finds a general solution for the d.e.  " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-k*x" "
6#,$*&%\"kG\"\"\"%\"xGF&!\"\"" }{TEXT -1 67 "  (the radioactive-decay \+
equation), where the unknown function is  " }{XPPEDIT 18 0 "x=x(t)" "6
#/%\"xG-F$6#%\"tG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "k" "6#%\"kG" 
}{TEXT -1 17 "  is a parameter:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(D(x)(t)=-k*x(t),x(t))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&%$_C1G\"\"\"-%$ex
pG6#,$*&%\"kGF*F'F*!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 22 "Note that inside the  " }{TEXT 264 6 "dsolve" }
{TEXT -1 27 "  command the derivative,  " }{XPPEDIT 18 0 "x" "6#%\"xG
" }{TEXT -1 26 "',  should be written as  " }{XPPEDIT 18 0 "D(x)" "6#-
%\"DG6#%\"xG" }{TEXT -1 38 "  and that the independent variable,  " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 52 ",  must be carried along (so
 that the d.e. becomes  " }{XPPEDIT 18 0 "D(x)(t)=-k*x(t)" "6#/--%\"DG
6#%\"xG6#%\"tG,$*&%\"kG\"\"\"-F(6#F*F.!\"\"" }{TEXT -1 14 "  instead o
f  " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-
k*x" "6#,$*&%\"kG\"\"\"%\"xGF&!\"\"" }{TEXT -1 26 " ). In Maple's resp
onse,  " }{XPPEDIT 18 0 "_C1" "6#%$_C1G" }{TEXT -1 84 "  denotes an ar
bitrary constant. (In general, Maple renders arbitrary constants as  \+
" }{XPPEDIT 18 0 "_C1" "6#%$_C1G" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "_C
2" "6#%$_C2G" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "_C3" "6#%$_C3G" }
{TEXT -1 87 ",  and so forth.) In standard calculus notation, we would
 write the solution formula as" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "x(t)=c*exp(-k*t)" "6#/-%\"xG6#%\"tG*&%\"cG\"\"\"-%$expG
6#,$*&%\"kGF*F'F*!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
59 "The next command produces a general solution for the d.e.  " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 "'' + " }{XPPEDIT 18 0 "a^2*x=
0" "6#/*&%\"aG\"\"#%\"xG\"\"\"\"\"!" }{TEXT -1 37 "  (the elastic-spri
ng equation with  " }{XPPEDIT 18 0 "a^2=k/m" "6#/*$%\"aG\"\"#*&%\"kG\"
\"\"%\"mG!\"\"" }{TEXT -1 41 " ), where the unknown function is again \+
 " }{XPPEDIT 18 0 "x=x(t)" "6#/%\"xG-F$6#%\"tG" }{TEXT -1 7 "  and  " 
}{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 17 "  is a parameter:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ds
olve(D(D(x))(t)+a^2*x(t),x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"tG,&*&%$_C1G\"\"\"-%$sinG6#*&%\"aGF+F'F+F+F+*&%$_C2GF+-%$cosG
F.F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
34 "Note that the second derivative,  " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 27 "'',  should be written as  " }{XPPEDIT 18 0 "D(D(x))" "6#
-%\"DG6#-F$6#%\"xG" }{TEXT -1 91 "  and that the independent variable \+
must be carried along, just as in the previous example." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "With a minor modific
ation, we can use the  " }{TEXT 287 6 "dsolve" }{TEXT -1 72 "  command
 to find symbolic solutions of i.v.p.s;  the modified format is" }}
{PARA 256 "" 0 "" {TEXT 265 50 "dsolve(\{DE,initial_condition(s)\},unk
nown_function)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 115 "Note t
he \"curly braces\" around the d.e. and initial condition(s). For exam
ple, let's solve the first-order i.v.p.  " }{XPPEDIT 18 0 "x" "6#%\"xG
" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-k*x" "6#,$*&%\"kG\"\"\"%\"xGF&!
\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "x(0)=x0" "6#/-%\"xG6#\"\"!%#x
0G" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "dsolve(\{D(x)(t)=-k*x(t),x(0)=x0\},x(t));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&%#x0G\"\"\"-%$expG6#
,$*&%\"kGF*F'F*!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 40 "Next, we solve the second-order i.v.p.  " }{XPPEDIT 
18 0 "x" "6#%\"xG" }{TEXT -1 5 "'' + " }{XPPEDIT 18 0 "a^2*x=0" "6#/*&
%\"aG\"\"#%\"xG\"\"\"\"\"!" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "x(0)=x0
" "6#/-%\"xG6#\"\"!%#x0G" }{TEXT -1 12 ",   x'(0) = " }{XPPEDIT 18 0 "
v0" "6#%#v0G" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dsolve(\{D(D(x))(t)+a^2*x(t)
=0,x(0)=x0,D(x)(0)=v0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"tG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF,F'F,F,F,F1!\"\"F,*&%#x0G
F,-%$cosGF/F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "In order to " }{TEXT 266 4 "plot" }{TEXT -1 30 " a soluti
on generated by the  " }{TEXT 267 6 "dsolve" }{TEXT -1 50 "  command, \+
we first have to give it a name, say,  " }{XPPEDIT 18 0 "s" "6#%\"sG" 
}{TEXT -1 30 ". The following command sets  " }{XPPEDIT 18 0 "s" "6#%
\"sG" }{TEXT -1 77 "  equal to the right-hand side of the immediately \+
preceding solution formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "s:=rhs(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF)%\"tGF)F)F)F.!
\"\"F)*&%#x0GF)-%$cosGF,F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 53 "Note:  The percent (%) is just a placehol
der for the " }{TEXT 268 61 "last output generated before execution of
 the present command" }{TEXT -1 84 " (which, in this case, happens to \+
be the solution formula produced by the preceding " }{TEXT 269 6 "dsol
ve" }{TEXT -1 74 " command). Instead of using the percent, we could ha
ve given a name (say, " }{XPPEDIT 18 0 "sol" "6#%$solG" }{TEXT -1 53 "
 ) to the solution formula and used that name in the " }{TEXT 270 3 "r
hs" }{TEXT -1 9 " command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol:=dsolve(\{D(D(x))(t)+a^2*x(t)=0
,x(0)=x0,D(x)(0)=v0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sol
G/-%\"xG6#%\"tG,&*&*&%#v0G\"\"\"-%$sinG6#*&%\"aGF.F)F.F.F.F3!\"\"F.*&%
#x0GF.-%$cosGF1F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s:=r
hs(sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*&*&%#v0G\"\"\"-%$
sinG6#*&%\"aGF)%\"tGF)F)F)F.!\"\"F)*&%#x0GF)-%$cosGF,F)F)" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "If we now specify
 the value of the parameter  " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 
26 "  and the initial values  " }{XPPEDIT 18 0 "x0" "6#%#x0G" }{TEXT 
-1 7 "  and  " }{XPPEDIT 18 0 "v0" "6#%#v0G" }{TEXT -1 16 ",  we can p
lot  " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 51 "  just like any othe
r function. Set, for example,  " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"
" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "x0=2" "6#/%#x0G\"\"#" }{TEXT -1 
9 " ,  and  " }{XPPEDIT 18 0 "v0=-3" "6#/%#v0G,$\"\"$!\"\"" }{TEXT -1 
2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "a:=1;  x0:=2;  v0:=-3;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G!\"$" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "In this case, the solution of t
he i.v.p. is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#%
\"tG!\"$*&\"\"#\"\"\"-%$cosGF&F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 45 "and a graph can be generated with the us
ual  " }{TEXT 271 4 "plot" }{TEXT -1 10 "  command:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(s,t=-
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oW[G:O3n=F-7$$\"3?+++b!)[/>Fd\\n$\"3EcRAHWrz8F-7$$\"3=+DJ!=_'>>Fd\\n$
\"3\"G8K_j$z1')FL7$$\"3;+]i0j\"[$>Fd\\n$\"3x;/0A\"3*=KFL7$$\"37](=#HA6
^>Fd\\n$!3=xWd3x)*\\EFL7$$\"33+D\"G:3u'>Fd\\n$!3'RSI<#om[%)FL7$$\"3/]i
SwSq$)>Fd\\n$!3o]M>j\"[BS\"F-7$$\"#?F*$!3(*)4c&G;nA>F--%'COLOURG6&%$RG
BG$\"#5!\"\"$F*F*F`do-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F(Feco%(DE
FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "T
o obtain the solution and its graph for other values of the parameter \+
 " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 27 "  or other initial value
s  " }{XPPEDIT 18 0 "x0" "6#%#x0G" }{TEXT -1 7 "  and  " }{XPPEDIT 18 
0 "v0" "6#%#v0G" }{TEXT -1 188 ",  just re-execute the last three comm
ands, changing only the first one as needed. For further details on pl
otting, especially on simultaneous plotting of several graphs, see the
 handout \"" }{HYPERLNK 17 "More on Basic Maple Commands" 1 "more.mws
" "" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 273 53 "2. Plotting Direction Fields and Numerical Solutions.
" }{TEXT -1 115 "  The basic Maple command for the graphical represent
ation of direction fields and approximate solution curves is  " }
{TEXT 274 6 "DEplot" }{TEXT -1 48 " ;  it is part of a package of rout
ines called  " }{TEXT 275 7 "DEtools" }{TEXT -1 51 ",  which must be e
xplicitly loaded with the command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "To plot a " }
{TEXT 276 15 "direction field" }{TEXT -1 37 " for a first-order d.e. o
f the form  " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }
{XPPEDIT 18 0 "f(t,x)" "6#-%\"fG6$%\"tG%\"xG" }{TEXT -1 21 ",  in the \+
rectangle  " }{XPPEDIT 18 0 "t=a..b" "6#/%\"tG;%\"aG%\"bG" }{TEXT -1 
3 ",  " }{XPPEDIT 18 0 "x=c..d" "6#/%\"xG;%\"cG%\"dG" }{TEXT -1 12 ", \+
 use the  " }{TEXT 277 6 "DEplot" }{TEXT -1 21 "  command in the form
" }}{PARA 256 "" 0 "" {TEXT 278 44 "DEplot(D(x)(t)=f(t,x(t)),x(t),t=a.
.b,x=c..d)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 "The followi
ng command, for example, generates a direction field for  " }{XPPEDIT 
18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG
\"\"#" }{TEXT -1 20 "  in the rectangle  " }{XPPEDIT 18 0 "t=-3..3" "6
#/%\"tG;,$\"\"$!\"\"F'" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x=-6..6" "6#
/%\"xG;,$\"\"'!\"\"F'" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "DEplot(D(x)(t)=x(t)^2,x(
t),t=-3..3,x=-6..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 295 295 295 
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "In addition to a dir
ection field, Maple will compute and display one or more " }{TEXT 288 
27 "approximate solution curves" }{TEXT -1 73 " of a given d.e. if sui
table initial conditions are specified in the form" }}{PARA 256 "" 0 "
" {TEXT 283 38 "\{[x(t0)=x0],[x(t1)=x1],[x(t2)=x2],...\}" }{TEXT -1 1 
"," }}{PARA 0 "" 0 "" {TEXT -1 55 "immediately following the range spe
cifications in the  " }{TEXT 284 6 "DEplot" }{TEXT -1 225 "  command. \+
Note that the set of initial conditions is enclosed in curly braces, w
hile each individual condition appears in brackets. Both, brackets and
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ven." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "F
or example, the following command produces a direction field for  " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "x^2" "6
#*$%\"xG\"\"#" }{TEXT -1 109 ",  as before, along with the (approximat
e) graph of a particular solution, satisfying the initial condition  \+
" }{XPPEDIT 18 0 "x(0)=1" "6#/-%\"xG6#\"\"!\"\"\"" }{TEXT -1 2 " :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "DEplot(D(x)(t)=x(t)^2,x(t),t=-3..3,x=-6..6,\{[x(0)=1]\});" }}
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0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 79 "In general, the given initial points do not have to \+
lie inside the plot window." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 70 "If initial conditions are given, specification \+
of the vertical range (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 35 "-r
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"  will choose the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 182 "-rang
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s sometimes useful, but not always. Let's try it with the above exampl
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