{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 255 0 0 1 2 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 0 1 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 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "New Page" -1 256 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 1 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 4 258 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 4 259 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 4 260 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 4 261 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 4 262 1 {CSTYLE "" -1 -1 "" 1 12 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 257 13 "homework7.mws" }}}{SECT 0 {PARA 257 "" 0 "" {TEXT -1 4 "Gett" }{TEXT 256 0 "" }{TEXT -1 11 "ing started" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT -1 11 "Section 3 .1" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 10" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode1 := diff( x(t), t ) = 2*x(t) + y(t) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode2 := diff( y(t), t ) = x(t) + y( t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "WINDOW := x = -3 .. 3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "DIRplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " title = `Direction Field f or #10` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "DIRplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IC := seq( seq( [x(0)=i/2, y(0)=j/2], j=-2..2 ), i=-2 ..2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "PHASEplot := DEpl ot( MODEL, VAR, DOMAIN, [ IC ], WINDOW," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " arrows=NONE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " scene=[ x, y ], linecolor=BLUE, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " title = `P hase Portrait for # 10` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASE plot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IC := [ x(0)=1, y(0)=1 ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P1:= DEplot( MODEL, VAR, DOMAIN, [ \+ IC ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x \+ ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P2:= DEplot( MODEL, VAR, DOMAIN, [ IC ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, linecolor=G REEN ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display( [ P1, P 2 ] , title = `x and y solution curves for # 10` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 11 "Section 3.2" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 1" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "c) Direction Field" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode1 := diff( x(t), t ) = 3*x(t) + 2*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode2 := diff( y(t), t ) = - 2*y(t) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 ] ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 5 ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "WINDOW := x = -3 .. 3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "IC := [ [x(0)= 0.03,y(0)=0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " [x (0)=-0.03,y(0)=0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " [x(0) =-1,y(0)= 2.5]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " [x(0)= 1 ,y(0)=-2.5] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, IC, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " title = `Direction F ield & Straight-Line Solutions for #1` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 10" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2, 2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " [ [ -1 , -2 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " [ 1 , -4 ] ] );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "a) Ei genvalues" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvalues( A \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "b) Eigenvectors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eigenvectors( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "c) Directio n Field" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( x(t ), t ) = -x(t) - 2*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode2 : = diff( y(t), t ) = x(t) - 4*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN \+ := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "WINDOW : = x = -3 .. 3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 \+ .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "IC := [ [x(0)= 3,y(0)= 3]," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 " [x(0)=-3,y(0)=-3]," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 28 " [x(0)= 3,y(0)= 1.5]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " [x(0)=-3,y(0)=-1.5] ];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, WINDO W, IC, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " \+ title = `Direction Field & Straight-Line Solutions for #1 0` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 19" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2 , 2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " [ [ 0 , 1 ]," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " [ -q , -p ] ] );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 28 "b) Characteristic polynomial" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "charpoly( A, lambda );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "c) Eigenv alues" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvalues( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 21" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2, 2," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " [ [ 0 , 1 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " [ 10 , -3 ] ] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 31 "b) Eigenvalues and eigenvectors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvalues( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eigenve ctors( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 11 "Section 3 .3" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 1" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Phase Portrait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode1 := diff( x(t), t ) = 3*x(t) + 2*y(t) :" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "ode2 := diff( y(t), t ) = - 2*y(t) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 ] ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 5 ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "WINDOW := x = -3 .. 3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "IC := [ seq( seq( [x(0)=i/5,y(0)= j], i=-10..10 ), j=[-3,3] ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, IC, \+ arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " \+ title = `Phase Portrait for #1` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 8" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Phase Portrait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( x(t), t ) = -x(t) - 2*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode2 := diff( y(t), t ) = x(t) - 4*y(t) \+ :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 ] ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 5 ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "WINDOW := x = -3 .. 3," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "IC := [ seq( seq( [x(0)=i/3,y(0)=j ], i=-9..9 ), j=[-3,3] )," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 " seq( seq( [x(0)=i, y(0)=j/3], i=[-3,3 ] ), j=-9..9 ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, \+ VAR, DOMAIN, WINDOW, IC, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " title = `Phase Portrait for #8` \+ ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 " # 13" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Phase Portrait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ode1 := diff( x(t), t ) = \+ y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ode2 := diff( y(t), t ) = 10*x(t) - 3*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL : = [ ode1, ode2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR : = \{ x(t), y(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 5 ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "WINDOW := x = -3 .. 3," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -3 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "IC := [ seq( seq( [x(0)=i/3,y(0)=j ], i=-9..9 ), j=[ -3,3] ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, VAR, DO MAIN, WINDOW, IC, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " title = `Phase Portrait for #13` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 261 "" 0 "" {TEXT -1 11 "Section 3.4" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2, 2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " [ \+ [ 2 , 2 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " [ -4 , 6 ] ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "a) Eigenvalues" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "eigenvalues( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 "e) Phase Pl ane and x- and y-solutions for the IVP" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ode1 := diff( x(t), t ) = 2*x(t) + 2*y(t) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ode2 := diff( y(t), t ) = -4*x(t) + 6*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode2 \+ ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "WINDOW := x = -10 .. 10," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " y = -10 .. 10 ;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " IC := [ [x(0)= 1,y(0)= 1] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, IC, arrows = ME DIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " title = `Phase Portrait for #4` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PH ASEplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "DOMAIN := t = 0 .. 2.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "P1:= DEplot( MODEL, VAR, DOMAIN, IC ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], a rrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "P2 := DEplot( MODEL, VAR, DOMAIN, IC," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, linecolor=GREEN ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "display( [ P1, P2 ] , title \+ = `x and y solution curves for # 4` );" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 10" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2, 2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " [ [ 2 , 2 ]," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 " [ -4 , 6 ] ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "a) General Solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lambda := 'lambda':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " eigenvalues( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eigenvectors( A );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Case 2: Complex Eigenvalues" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "alpha := 4 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "be ta := 2 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lambda := alpha + I * \+ beta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "V := vector( 2, [ 1 , 1+I \+ ] ); # be careful to distinguish between Maple's 1 and I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "The linearly independent solutions are found by the usual step s: exp( eigenvalue * t ) * eigenvector, but recall that the two soluti ons are the real and imaginary parts of this expression. These calcula tions are messy - by hand. With Maple it's a little better, but there \+ are some extra commands that must be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 304 "We know that the two solutions will consist of the product of an exponential formed with the real pa rt of the eigenvalues and a vector with trigonometric terms formed usi ng the imaginary part of the eigenvalues. Let's begin by looking at th e part of the solution that leads to complex-valued expressions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume( t, real );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "mess := exp( I* beta *t ) * \+ evalm( V );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "mess2 := con vert( evalc( evalm( mess ) ), trig );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "t := 't';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "messRE := map( Re, mess2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "messIM := map( Im, mess2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Using the re al and imaginary parts we can construct the two linearly independent s olutions for this system:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Y1 := exp( alpha *t) * convert( evalm( messRE ), matrix );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Y2 := exp( alpha *t) * conve rt( evalm( messIM ), matrix );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The following two-parameter family of sol utions is the general solution to the system of ODEs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "SOLgen := c1 * Y1 + c2 * Y2;" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "c) Explicit x- and y-solution curves" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot( [ exp(4*t)*cos(2*t), \+ exp(4*t)*(cos(2*t)-sin(2*t)) ], t=0..2.5, color=[BLUE,GREEN] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 262 "" 0 "" {TEXT -1 11 "Sectio n 3.5" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 1" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( x(t), t ) = -3*x(t) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ode2 := diff( y(t), t ) = x(t) - 3 * y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := [ ode1, ode 2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y( t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "c) Direction Field" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "WINDOW := x = -1 .. 1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -1 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "DIRplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, arrows =THIN, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " \+ title = `Direction Field for #1` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "DIRplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "d) Phase Plane" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "WINDOW := x = -1 .. 1," }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 23 " y = -1 .. 1 ;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "IC := [ seq( seq( [x(0)=i, y(0)=j/3 ], j=-3 ..3 ), i=[-1,1] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " seq( \+ seq( [x(0)=i/3,y(0)=j ], j=[-1,1] ), i=-3..3 ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, \+ WINDOW, IC, arrows=THIN, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " title = `Phase Plane for #1` ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 39 "e) x- and y-solution curves for the IVP" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "IC := [ x(0) = 1, y(0) = 0 ];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC ]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows =NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P2:= DE plot( MODEL, VAR, DOMAIN, [ IC ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, linecolor=GREEN ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "display( [ P1, P2 ] , title \+ = `x and y solution curves for # 1` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "# 4" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 35 "c) x- and y-solution curves for IVP" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "plot( [ exp(-3*t), t*exp(-3*t) ], t = 0 .. 3, \+ color=[BLUE,GREEN] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "# 18" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix( 2, 2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " [ [ 2, 4 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " [ 3, 6 ] ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "a) eigenvalues" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvalues( A );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 15 "b) eigenvectors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eigenvectors( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "c) Phase Plane" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( x(t), t ) = 2*x(t) + \+ 4*y(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode2 := diff( y(t), t ) \+ = 3*x(t) + 6*y(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "MODEL := \{ o de1, ode2 \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "VAR := \{ \+ x(t), y(t) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "DOMAIN := t = -1 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "WINDOW := x = -1 .. 1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " y = -1 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "IC := [ seq( seq( [x(0)=i, y(0)=j/3 ], j=-3..3 ), i= [-1,1] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " seq( seq( [x(0 )=i/3,y(0)=j ], j=[-1,1] ), i=-3..3 ) ];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "PHASEplot := DEplot( MODEL, VAR, DOMAIN, WINDOW, IC , arrows=THIN, arrows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " title = `Phase Plane for #18` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 39 "d) x- and y -solution curves for the IVP" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "IC := [ x(0) = 1, y(0) = 0 ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC ]," }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows=NONE, li necolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P2:= DEplot( MOD EL, VAR, DOMAIN, [ IC ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " \+ scene=[ t, y ], arrows=NONE, linecolor=GREEN ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display( [ P1, P2 ] , title = `x an d y solution curves for # 18` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 }