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{SECT 0 {EXCHG {PARA 218 "" 0 "" {TEXT 268 63 "Matrices, Eigenvectors,
 Eigenvalues, and Differential Equations" }}}{SECT 1 {PARA 219 "" 0 ""
 {TEXT 269 37 "1: Eigenvalues and Eigenvectors: The " }{TEXT 270 9 "Ei
genvals" }{TEXT 269 8 " command" }}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 8 "restart;" }}}{PARA 220 "" 0 "" }{PARA 220 "" 0 "" 
{TEXT 272 30 "Consider the 2 x 2 matrix A = " }{XPPEDIT 18 0 "matrix([
[1, 1], [4, 1]]);" "6#-%'matrixG6#7$7$\"\"\"F(7$\"\"%F(" }{TEXT 273 1 
" " }{TEXT 273 1 " " }{TEXT 273 1 " " }}{PARA 220 "" 0 "" {TEXT 272 
37 "We define a matrix in Maple with the " }{TEXT 274 6 "matrix" }
{TEXT 272 8 " command" }}{PARA 220 "" 0 "" {TEXT 272 9 "syntax:  " }
{TEXT 274 7 "matrix(" }{TEXT 272 9 "# of rows" }{TEXT 274 2 ", " }
{TEXT 272 12 "# of columns" }{TEXT 274 2 ",[" }{TEXT 272 23 "list of m
atrix elements" }{TEXT 274 2 "])" }}{PARA 220 "" 0 "" {TEXT 272 49 "Fo
r example, the matrix above would be entered by" }}{EXCHG {PARA 220 ">
 " 0 "" {MPLTEXT 1 271 27 "A := matrix(2,2,[1,1,4,1]);" }}{PARA 221 ""
 1 "" {XPPMATH 20 "6#>I\"AG6\"-I'matrixG6$I*protectedGF)I(_syslibGF%6#
7$7$\"\"\"F.7$\"\"%F." }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" 
{TEXT 272 134 "First, it may be worth checking to see if the matrix is
 singular. We can do this by evaluating the determinent of the matrix \+
with the " }{TEXT 274 3 "det" }{TEXT 272 22 " command found in the " }
{TEXT 274 6 "linalg" }{TEXT 272 35 " library. First we load the librar
y" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 13 "with(linalg):" }}
{PARA 222 "" 1 "" {TEXT 276 79 "Warning, the protected names norm and \+
trace have been redefined and unprotected" }{TEXT 276 1 "\n" }}}
{EXCHG {PARA 220 "" 0 "" {TEXT 272 33 "then check the determinent of A
. " }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 7 "det(A);" }}{PARA 
221 "" 1 "" {XPPMATH 20 "6#!\"$" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 
"" 0 "" {TEXT 272 61 "Since the determinent is nonzero, the matrix is \+
nonsingular. " }}{PARA 220 "" 0 "" {TEXT 272 258 "We can find two line
arly independent solutions of the matrix differential equation y' = Ay
, if we can find the eigenvalues and 2 linearly independent eigenvecto
rs of the matrix A. This is handled in a rather odd fashion with Maple
. The command to do this is " }{TEXT 274 9 "Eigenvals" }{TEXT 272 41 "
, but the result must be evaluated using " }{TEXT 274 5 "evalf" }
{TEXT 272 120 ". You must also input an unassigned variable (in the be
low example called \"vecs\") in which to hold the eigenvectors.    " }
}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 36 "eigvals := evalf(Eigen
vals(A,vecs));" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(eigvalsG6\"-I've
ctorG6$I*protectedGF)I(_syslibGF%6#7$$\"+++++I!\"*$!+++++5F/" }{TEXT 
275 1 " " }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 38 "We can access each
 of these values by " }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 22 
"eigval1 := eigvals[1];" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(eigval1
G6\"$\"+++++I!\"*" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 22 "eigval2 := eigvals[2];" }}{PARA 221 "" 1 "" 
{XPPMATH 20 "6#>I(eigval2G6\"$!+++++5!\"*" }{TEXT 275 1 " " }}}{EXCHG 
{PARA 220 "" 0 "" {TEXT 272 117 "Clearly we have two different eigenva
lues so the eigenvectors are linearly independent.  To view these eige
nvectors: " }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 12 "print(vec
s);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#-I'matrixG6$I*protectedGF&I(_s
yslibG6\"6#7$7$$\"+bf8sW!#5$!+W*p,f&F.7$$\"+5>FW*)F.$\"+*)R.=6!\"*" }
{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 79 "We can acces
s the first column of this matrix (the first eigenvector) with the " }
{TEXT 274 3 "col" }{TEXT 272 36 " command also in the linalg library."
 }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 13 "with(linalg):" }}}
{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 23 "eigvec1 := col(vecs,1);"
 }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(eigvec1G6\"-I'vectorG6$I*protec
tedGF)I(_syslibGF%6#7$$\"+bf8sW!#5$\"+5>FW*)F/" }{TEXT 275 1 " " }}}
{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 23 "eigvec2 := col(vecs,2);"
 }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(eigvec2G6\"-I'vectorG6$I*protec
tedGF)I(_syslibGF%6#7$$!+W*p,f&!#5$\"+*)R.=6!\"*" }{TEXT 275 1 " " }}}
{EXCHG {PARA 220 "" 0 "" {TEXT 272 220 "Keep in mind that eigenvectors
 are unique only up to a constant multiple so Maple results may not im
mediately look like the eigenvectors calculated by hand. Notice that t
he eigenvectors are clearly linearly independent. " }}{PARA 220 "" 0 "
" {TEXT 272 13 "Check that A " }{XPPEDIT 18 0 "xi[i];" "6#&%#xiG6#%\"i
G" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 4 "  \+
= " }{XPPEDIT 18 0 "lambda[i];" "6#&%'lambdaG6#%\"iG" }{TEXT 273 1 " "
 }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 2 "  " }{XPPEDIT 18 0 "x
i[i]" "6#&%#xiG6#%\"iG" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 273 
1 " " }{TEXT 272 10 "   or   A " }{XPPEDIT 18 0 "xi[i];" "6#&%#xiG6#%
\"iG" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 4 
"  - " }{XPPEDIT 18 0 "lambda[i];" "6#&%'lambdaG6#%\"iG" }{TEXT 273 1 
" " }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 2 "  " }{XPPEDIT 18 
0 "xi[i]" "6#&%#xiG6#%\"iG" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 
273 1 " " }{TEXT 272 7 " = 0.  " }}{PARA 220 "" 0 "" {TEXT 272 34 "Mul
tiplying matrices requires the " }{TEXT 274 8 "multiply" }{TEXT 272 
22 " command found in the " }{TEXT 274 6 "linalg" }{TEXT 272 59 " libr
ary and multiplying a vector by a scalar requires the " }{TEXT 274 9 "
scalarmul" }{TEXT 272 27 " command also found in the " }{TEXT 274 6 "l
inalg" }{TEXT 272 10 " library. " }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 13 "with(linalg):" }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 49 "multiply(A,eigvec1) - scalarmul(eigvec1,eigval1);" 
}}{PARA 221 "" 1 "" {XPPMATH 20 "6#,&-I'vectorG6$I*protectedGF'I(_sysl
ibG6\"6#7$$\"+'yS;M\"!\"*$\"+t:G$o#F.\"\"\"F$!\"\"" }{TEXT 275 1 " " }
}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 49 "multiply(A,eigvec2), -
scalarmul(eigvec2,eigval2);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6$-I'vec
torG6$I*protectedGF&I(_syslibG6\"6#7$$\"+Y*p,f&!#5$!+*)R.=6!\"*,$-F$6#
7$$\"+W*p,f&F-F.!\"\"" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" 
{TEXT 272 72 "This second one is close to [0,0] but not exactly (round
-off error !!). " }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 16 "# e
nd of section" }}}}{SECT 1 {PARA 219 "" 0 "" {TEXT 269 38 "2: Matrix D
ifferential Equations: The " }{TEXT 270 8 "matrixDE" }{TEXT 269 8 " co
mmand" }}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 8 "restart;" }}}
{PARA 220 "" 0 "" {TEXT 272 54 "Consider the 2 x 2 matrix differential
 equation  y' = " }{XPPEDIT 18 0 "matrix([[1, 1], [4, 1]]);" "6#-%'mat
rixG6#7$7$\"\"\"F(7$\"\"%F(" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 
273 1 " " }{TEXT 272 2 " y" }}{PARA 220 "" 0 "" {TEXT 272 27 "First we
 define the matrix " }}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 27 "A
 := matrix(2,2,[1,1,4,1]);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I\"AG6
\"-I'matrixG6$I*protectedGF)I(_syslibGF%6#7$7$\"\"\"F.7$\"\"%F." }
{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" {TEXT 274 38 "Solving matr
ix differential equations:" }{TEXT 272 200 " Here we will use a progra
m specifically designed for solving differential equations in the matr
ix form y' = A(t) y + B(t). This program is contained in a library cal
led DEtools so we load this libray" }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 14 "with(DEtools):" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 
272 13 "The command  " }{TEXT 274 16 "matrixDE(A,B, t)" }{TEXT 272 
334 " solves the system y' = Ay + B.  The result is a pair of solution
s. The first is a set of solutions to the homogeneous equation y' = Ay
 and the second is a particular solution of the nonhomogeneous equatio
ns y' = Ay + B. Since our example has B = 0, we can exclude it from th
e input.  I call the result gensol for 'general solution'. " }}}
{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 24 "gensol := matrixDE(A,t);
" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I'gensolG6\"7$-I'matrixG6$I*prot
ectedGF*I(_syslibGF%6#7$7$-I$expGF)6#,$I\"tGF%!\"\"-F06#,$F3\"\"$7$,$F
/!\"#,$F5\"\"#-I'vectorGF)6#7$\"\"!FB" }{TEXT 275 1 " " }}}{EXCHG 
{PARA 220 "" 0 "" {TEXT 272 60 "We are concerned with the first matrix
 here which I call sol" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 
17 "sol := gensol[1];" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I$solG6\"-I
'matrixG6$I*protectedGF)I(_syslibGF%6#7$7$-I$expGF(6#,$I\"tGF%!\"\"-F/
6#,$F2\"\"$7$,$F.!\"#,$F4\"\"#" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "
" 0 "" {TEXT 272 228 "Here, each column represents a solution.  Notice
,  Maple went back to the simpler form of the eigenvectors in this sol
ution. These differ from the eigenvectors found in section 1 but agree
 with the eigenvectors found by hand.   " }}}{EXCHG {PARA 220 "> " 0 "
" {MPLTEXT 1 271 15 "#end of section" }}}}{SECT 1 {PARA 219 "" 0 "" 
{TEXT 269 25 "3: Direction Fields: The " }{TEXT 270 10 "dfieldplot" }
{TEXT 269 8 " command" }}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 8 "
restart;" }}}{PARA 220 "" 0 "" {TEXT 272 34 "To get a direction field \+
in the  (" }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" }{TEXT 273 1 " " 
}{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 2 ", " }{XPPEDIT 18 0 "y[
2];" "6#&%\"yG6#\"\"#" }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 273 1 
" " }{TEXT 272 98 ")  phase plane from Maple we consider the two diffe
rential equations described by the system y' = " }{XPPEDIT 18 0 "matri
x([[1, 1], [4, 1]]);" "6#-%'matrixG6#7$7$\"\"\"F(7$\"\"%F(" }{TEXT 
273 1 " " }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 4 " y. " }}
{PARA 220 "" 0 "" {TEXT 272 11 "These are  " }{XPPEDIT 18 0 "diff(y[1]
,t) = y[1]+y[2];" "6#/-%%diffG6$&%\"yG6#\"\"\"%\"tG,&F'F*&F(6#\"\"#F*"
 }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 10 "   \+
 and   " }{XPPEDIT 18 0 "diff(y[2],t) = 4*y[1]+y[2];" "6#/-%%diffG6$&%
\"yG6#\"\"#%\"tG,&*&\"\"%\"\"\"&F(6#F/F/F/F'F/" }{TEXT 273 1 " " }
{TEXT 273 1 " " }{TEXT 273 1 " " }{TEXT 272 2 ". " }}{PARA 220 "" 0 ""
 {TEXT 272 38 "Now describe them in Maple language by" }}{EXCHG {PARA 
220 "> " 0 "" {MPLTEXT 1 271 41 "diffeq1 := diff(y1(t),t) = y1(t) + y2
(t);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(diffeq1G6\"/-I%diffGI*prot
ectedGF)6$-I#y1GF%6#I\"tGF%F.,&F+\"\"\"-I#y2GF%F-F0" }{TEXT 275 1 " " 
}}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 43 "diffeq2 := diff(y2(t)
,t) = 4*y1(t) + y2(t);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(diffeq2G
6\"/-I%diffGI*protectedGF)6$-I#y2GF%6#I\"tGF%F.,&-I#y1GF%F-\"\"%F+\"\"
\"" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 53 "The com
mand for plotting a direction field is in the " }{TEXT 274 7 "DEtools"
 }{TEXT 272 8 " library" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 
14 "with(DEtools):" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 41 "The comm
and to plot a direction field is " }{TEXT 274 10 "dfieldplot" }{TEXT 
272 18 ".  The syntax is: " }}{PARA 220 "" 0 "" {TEXT 274 12 "dfieldpl
ot([" }{TEXT 272 22 "differential equations" }{TEXT 274 4 "], [" }
{TEXT 272 19 "dependent variables" }{TEXT 274 3 "], " }{TEXT 272 7 "t \+
range" }{TEXT 274 2 ", " }{TEXT 272 8 "y1 range" }{TEXT 274 2 ", " }
{TEXT 272 8 "y2 range" }{TEXT 274 1 ")" }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 124 "dfieldplot([diffeq1,diffeq2], [y1(t),y2(t)],t=0..1
,y1=-3..3,y2=-3..3,        title=`Direction Field for y' = Ay`,color=r
ed);" }}{PARA 223 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%'CURV
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$\"3%4$Hf%R(*e9$F*7$$\"3aWc<ef'>o#F*$\"3c<Y[.[\"y4$F*7%7$$\"3Qe9*oOf8%
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HF*$\"3e#*4f$R\"y(4$F*-%*THICKNESSG6#\"\"\"-%&COLORG6&%$RGBG$\"#5!\"\"
$FacblFdgclFegcl-%&TITLEG6$Q<Direction~Field~for~y'~=~Ay6\"-%%FONTG6$%
*HELVETICAGFcgcl-%+AXESLABELSG6$Q#y1FjgclQ#y2FjgclF[hcl-%%VIEWG6$;$!2/
++++++I$!#;$\"2/++++++I$FjhclFghcl" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 
45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 51 "Make sur
e to include a range of t,y1 and y2 values." }}}{PARA 220 "" 0 "" }}
{SECT 1 {PARA 219 "" 0 "" {TEXT 269 39 "4: Phase Portraits and Time Pl
ots: The " }{TEXT 270 6 "DEplot" }{TEXT 269 8 " command" }}{EXCHG 
{PARA 220 "> " 0 "" {MPLTEXT 1 271 8 "restart;" }}}{EXCHG {PARA 220 ">
 " 0 "" {MPLTEXT 1 271 44 "diffeq1 := diff(y1(t),t) = -y1(t) -8* y2(t)
;" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(diffeq1G6\"/-I%diffGI*protect
edGF)6$-I#y1GF%6#I\"tGF%F.,&F+!\"\"-I#y2GF%F-!\")" }{TEXT 275 1 " " }}
}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 45 "diffeq2 := diff(y2(t),t
) = 6*y1(t) - 1*y2(t);" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I(diffeq2G
6\"/-I%diffGI*protectedGF)6$-I#y2GF%6#I\"tGF%F.,&-I#y1GF%F-\"\"'F+!\"
\"" }{TEXT 275 1 " " }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 52 "The com
mand for plotting direction fields is in the " }{TEXT 274 7 "DEtools" 
}{TEXT 272 8 " library" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 271 
14 "with(DEtools):" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 62 "The comm
and to plot phase portraits (and direction fields) is " }{TEXT 274 6 "
DEplot" }{TEXT 272 18 ".  The syntax is: " }}{PARA 220 "" 0 "" {TEXT 
274 94 "DEplot([differential equations], [dep vars], t range, y1 range
, y2 range, [list of init cond])" }}}{EXCHG {PARA 220 "> " 0 "" 
{MPLTEXT 1 271 234 "DEplot([diffeq1,diffeq2], [y1(t),y2(t)],t=-10..10,
y1=-3..3,y2=-3..3,[[y1(0)=0,y2(0)=1],[y1(0)=-1,y2(0)=1],[y1(0)=1.1,y2(
0)=-2],[y1(0)=0,y2(0)=-1]],stepsize=.01,        title=`Phase Portrait \+
and Direction Field for y' = Ay`,color=red);" }}{PARA 223 "" 1 "" 
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$$!3\\l/'e[ab$)*Fijcm$!3o7Qw5@='p%F^]s7$$!3WJb5;,u(*fFijcm$!3Q=6/I&zmp
%F^]s7$$!3;AA\")*RMo?#Fijcm$!3S%f%*3H\"RuYF^]s7$$\"3Se7h9\"3(>:Fijcm$!
3Lia^im()HYF^]s7$$\"3#=R-4i!4l^Fijcm$!3)>97Ki\"zjXF^]s7$$\"3l#))fU[UKr
)Fijcm$!3')4%zVw$)oZ%F^]s-%&STYLEG6#%%LINEG-Fiecl6#\"\"$-%&TITLEG6$QOP
hase~Portrait~and~Direction~Field~for~y'~=~Ay6\"-%%FONTG6$%*HELVETICAG
Fafcl-%+AXESLABELSG6$Q#y1FbfiqQ#y2FbfiqFcfiq-%%VIEWG6$;$!2/++++++I$!#;
$\"2/++++++I$FbgiqF_giq-F]fcl6&F_fcl$\"1#=t:!zg>!*FbgiqFggiq$\"2CK-)H+
++?F*" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}
}}{EXCHG {PARA 220 "" 0 "" {TEXT 272 153 "Since there were 4 sets of i
nitial conditions, 4 solution trajectories are displayed.  More trajec
tories can be displayed using more initial conditions. " }}}{EXCHG 
{PARA 224 "> " 0 "" {MPLTEXT 1 277 20 "y1o := 2: y2o := -2:" }}}
{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 277 56 "odesystem := \{diffeq1, \+
diffeq2, y1(0)=y1o, y2(0) = y2o\};" }}{PARA 221 "" 1 "" {XPPMATH 20 "6
#>I*odesystemG6\"<&/-I#y2GF%6#\"\"!!\"#/-I%diffGI*protectedGF06$-F)6#I
\"tGF%F4,&-I#y1GF%F3\"\"'F2!\"\"/-F/6$F6F4,&F6F9F2!\")/-F7F*\"\"#" }
{TEXT 275 1 " " }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 277 32 "sol :
= dsolve(odesystem,numeric," }{MPLTEXT 1 277 20 "output=listprocedure"
 }{MPLTEXT 1 277 2 ");" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I$solG6\"7
%/I\"tGF%f*6#F(6%I$resGF%I%dataGF%I)solnprocGF%6#IaoCopyright~(c)~1993
~by~the~University~of~Waterloo.~All~rights~reserved.GF%F%C&>8%`6$I$Get
GF%I$SetGF%b6#I+thismoduleGF%6#I%DataGF%F%F5F%F%F%F%I+_m24167188GF%F;>
8&-_F3F66#Q/soln_procedureF%@&-I'memberGI*protectedGFG6$9$7$Q%lastF%.I
%lastGF%C$>8$-F?6#FK@$-I%typeGFG6$FP.I%listGFGO&FP6#\"\"\"3-FU6$FII\"=
GFG-FF6$-I$lhsGFG6#FI7$Q(initialF%.I(initialGF%C$@%-FU6$-I$rhsGFGF_oFW
>FP-F?6#/Fao7$\"\"!-I#opGFG6#Fho>FP-F?6#/Fao7$FfnFhoFSFIF%F%F%/-I#y1GF
%F*f+F*6'F,F-F.I)outpointGF%I&y1(t)GF%6#IinCopyright~(c)~2000~by~Water
loo~Maple~Inc.~All~rights~reserved.GF%F%C)>I8_EnvDSNumericSaveDigitsGF
%I'DigitsGF%>Fdq\"#9@%/I-_EnvInFsolveGF%I%trueGFG>8'-&I&evalfGFG6#FcqF
_o>F\\r-F_rF_oF2F>@$4-FU6$F\\r.I(numericGFGC$@.-FF6$FI7+Q&startF%.I&st
artGF%Q'methodF%.I'methodGF%Q%leftF%Q&rightF%Q)leftdataF%Q*rightdataF%
Q+enginedataF%C$@%-FF6$FI7$F_sFbs>FP-F?6#-I(convertGFG6$FI.I'stringGFG
>FP-F?F_o@&-FU6$FP.I&arrayGFGO-I%evalGFG6$FPFfn0FPQ)procnameF%OFP-FF6$
FI7$FKFaoC$Fft@$FTO&FP6#\"\"#-FF6$FI7$FLFboC$F^tFhu3-FU6$F\\rFjn-FF6$-
F^o6#F\\rF`oC$@%-FU6$-FioFgvFW>FP-F?6#/Fao7$F_p-Fap6#F\\w>FP-F?6#/Fao7
$F\\vF\\wFhu/F\\rQ.solnprocedureF%O-F_u6#F?/F\\rQ(sysvarsF%O7%F(Fip-I#
y2GF%F*@%09!I(unknownGF%O-.FfxF_oC$>8(-I(pointtoGFG6#&-FA6#Q0soln_proc
eduresF%F[vO-.F]yF_oZ%C$>FP-F?FgvFjuF%YF%F%F%F%F%/Fbxf+F*6'F,F-F.F]qI&
y2(t)GF%F_qF%C)FbqFeqFgqF2F>@$FdrC$@.F[sFisFduC$Fft@$FTO&FP6#\"\"$F]vC
$F^tFfzFavC$@%FjvF]w>FP-F?6#/Fao7$FjzF\\wFfzFiwF[xF^xF`x@%FexFhxC$>F]y
-F_y6#&FbyFizFeyZ%C$FjyFhzF%F\\zF%F%F%F%" }{TEXT 275 1 " " }}}{EXCHG 
{PARA 221 "" 1 "" {XPPMATH 20 "6#-I#y1G6\"6#I\"tGF%" }{TEXT 275 1 " " 
}}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 277 51 "y1sol := subs(sol,y1(
t)): y2sol := subs(sol,y2(t)):" }}}{EXCHG {PARA 224 "> " 0 "" 
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