{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }} {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 0 194 "dfieldplot([diff(x(t),t)=0.2*x(t)-0.5*y( t),\n diff(y(t),t)=0.1*x(t)-0.3*y(t)],\n [x(t),y (t)],t=-2..2,x=-0.1..0.1,y=-0.1..0.1,\n arrows=thin,title=` Saddle Surface`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "dfiel dplot([diff(x(t),t)= 11*y(t),\n diff(y(t),t)= -3*x(t)],\n \+ [x(t),y(t)],t=-2..2,x=-0.1..0.1,y=-0.1..0.1,\n ar rows=thin,title=`What???`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "dfieldplot([diff(x(t),t)= y(t),\n diff(y(t),t)= -0.1*x (t)-0.9*y(t)],\n [x(t),y(t)],t=-2..2,x=-0.3..0.3,y=-0.3..0. 3,\n arrows=thin,title=`what???`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "phaseportrait([D(x)(t)= y(t),\n D(y )(t)=-0.1*x(t)-0.9*y(t)], \\\n [x(t),y(t)],t=0..20,\n \+ [[x(0)=0.2,y(0)=0.2],[x(0)=-0.2,y(0)=-0.2]],\n steps ize=.1, \n scene=[x(t),y(t)],linecolor=blue,\n \+ method=classical[heunform],\n title=`Down the Drain?`);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "phaseportrait([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)],l inecolor=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 eigenvector s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A1:=matrix(2,2,[3, 4, 4, -5] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvectors(A1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "There are two eigenvalues. Th ey are real, distinct and negative. The fact that one is negative and one is positive tells you that some solutions to this system will dec ay to zero while others will grow exponentially. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "9 9 8 0" 136 } {VIEWOPTS 1 1 0 1 1 1803 }