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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Examples for Lecture 19, C99" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "filename:
  lecture19.mws" }}{PARA 0 "" 0 "" {TEXT -1 16 "updated: 2/24/99" }}
{PARA 0 "" 0 "" {TEXT -1 30 "author:   Arthur C. Heinricher" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The basic tools
 for working with differential equations are in the " }{TEXT 0 7 "DEto
ols" }{TEXT -1 23 " library.  So load it. " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "with(DEtools):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Example #1:  Saddle Surface" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }
{HYPERLNK 17 "dfieldplot" 2 "dfieldplot" "" }{TEXT -1 65 " command wil
l draw the direction field for a first-order system. " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 187 "dfieldplot([diff(x(t),t)=8*x(t)-11*y(t),
\n            diff(y(t),t)=6*x(t)-9*y(t)],\n            [x(t),y(t)],t=
-2..2,x=-0.3..0.3,y=-0.3..0.3,\n            arrows=thin,title=`Saddle \+
Surface`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "phaseportrai
t([D(x)(t)=8*x(t)-11*y(t),\n             D(y)(t)=6*x(t)-9*y(t)], \\\n \+
           [x(t),y(t)],t=0..0.7,\n            [[x(0)=2,y(0)=1.7],[x(0)
=2.0,y(0)=2.0]],\n             stepsize=.005, \n             scene=[x(
t),y(t)],linecolor=blue,\n             method=classical[heunform],\n  \+
           title=`Saddle Surface`);\n" }}}{PARA 4 "" 0 "" {TEXT -1 0 "
" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "A little linear algebra" }}
{PARA 0 "" 0 "" {TEXT -1 123 "Remember that solving a system of ODE's \+
is really a linear algebra problem:  you need to find eigenvalues and \+
eigenvectors." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=matrix(2,2,[8,-11,
6,-9]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "There ar
e two eigenvalues.  They are real, distinct and negative.  The fact th
at one is negative and one is positive tells you that some solutions t
o this system will decay to zero while others will grow exponentially.
  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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